Not sure if this is what you're interested in, but let me take up the question of what large cardinal axioms the existence of $h$ is known to be consistent with. The answer is: all large cardinal axioms that are known to have a canonical inner model. Beyond that, the consistency question remains open: indeed, it remains open whether large cardinal axioms beyond the reach of inner model theory could outright imply the existence of a precipitous ideal on $\omega_1$, and thereby refute the existence of $h$.

It seems plausible, however, that $h$ in fact exists in the canonical inner models yet to be discovered. This is because one can construct a function that is almost exactly like $h$ (see the fourth-to-last paragraph below) using a general condensation principle due to Woodin called Strong Condensation, which likely holds in all canonical inner models. If $\kappa$ is a cardinal, Strong Condensation at $\kappa$ states that there is a surjective function $f : \kappa\to H(\kappa)$ of such that for all $M\prec (H(\kappa),f)$, letting $(H_M,f_M)$ be the transitive collapse of $(M,f)$, $f_M = {f}\restriction H_M$.

All the known canonical inner models satisfy Strong Condensation at their least strong limit cardinal. (Strong condensation cannot hold at any cardinals past the first Ramsey cardinal.) Moreover by a theorem of Woodin, under $\text{AD}^+ + V = L(P(\mathbb R))$, for a Turing cone of reals $x$, $\text{HOD}_x$ satisfies Strong Condensation at its least strong limit cardinal. This heuristically argues that Strong Condensation should hold at the least strong limit cardinal in canonical inner models satisfying arbitrarily strong large cardinal axioms, assuming that such models exist. The reason is that the pattern observed in inner model theory to date suggests that these models should locally resemble the $\text{HOD}$s of determinacy models.

Here is the approximation to the existence of $h$ one gets assuming Strong Condensation at the least strong limit cardinal $\gamma$: There is a set $a\subseteq \omega_1$ and a function $g:\omega_1\to\omega_1$ such that for any $N\prec V_\gamma$ with $a\in N$, $g(N\cap \omega_1) > \text{ot}(N\cap \gamma)$. The rest of this answer consists of a proof of this fact.

Fix $f : \gamma\to H(\gamma)$ witnessing Strong Condensation at $\gamma$. The first step of the proof is cosmetic. One uses a theorem of Woodin which states that $f$ is definable over $H(\gamma)$ from the parameter $f \restriction \omega_1$. Let $a\subseteq \omega_1$ code $f\restriction \omega_1$. (It is an easy exercise to show that $f\restriction \omega_1\in H(\omega_2)$.) Then every $N\prec V_\gamma$ with $a\in N$ has the property that $N\cap H(\gamma)\prec (H(\gamma),f)$.

For $\alpha < \gamma$, let $P_\alpha = f[\alpha]$. Note that the $P_\alpha$ are increasing with union $H(\gamma)$, and if $M\prec (H(\gamma),f)$ and $\text{ot}(M\cap \gamma) = \alpha$, then the transitive collapse of $M$ is equal to $P_\alpha$. The structures $P_\alpha$ will play the role of the $L_\alpha$ hierarchy in Larson's proof.

For every $\xi < \omega_1$, let $g(\xi)$ be the least ordinal $\alpha$ such that there is a surjection from $\omega$ to $\xi$ in $P_\alpha$. Suppose $N\prec V_\gamma$ and $a\in N$. We will show that $g(N\cap \omega_1) > \text{ot}(N\cap \gamma)$. Let $M = N\cap H(\gamma)$, so $M\prec (H(\gamma),<)$. Clearly it suffices to show that $g(M\cap \omega_1) > \text{ot}(M\cap \gamma)$. Let $H_M$ be the transitive collapse of $M$. Then $M\cap \omega_1 = \omega_1^{H_M}$ and letting $\beta = \text{ot}(M\cap \gamma)$, $H_M = P_\beta$. Assume towards a contradiction that $\beta \geq g(\omega_1^{H_M})$. By the definition of $g$, there is a surjection from $\omega$ to $\omega_1^{H_M}$ in $P_\beta$. But since $P_\beta = H_M$, this contradicts that $\omega_1^{H_M}$ is uncountable in $H_M$.

Not sure if this is what you're interested in, but let me take up the question of what large cardinal axioms are known to be consistent with the existence of $h$. The answer is all large cardinal axioms known to have a canonical inner model. Beyond that, the consistency question remains open: indeed, it remains open whether large cardinal axioms beyond the reach of inner model theory could outright imply the existence of a precipitous ideal on $\omega_1$, and thereby refute the existence of $h$.

It seems plausible, however, that $h$ in fact exists in the canonical inner models yet to be discovered. This is because one can construct a function that is almost exactly like $h$ (see the fourth-to-last paragraph below) using a general condensation principle due to Woodin called Strong Condensation, which likely holds in all canonical inner models. If $\kappa$ is a cardinal, Strong Condensation at $\kappa$ states that there is a surjective function $f : \kappa\to H(\kappa)$ of such that for all $M\prec (H(\kappa),f)$, letting $(H_M,f_M)$ be the transitive collapse of $(M,f)$, $f_M = {f}\restriction H_M$.

All the known canonical inner models satisfy Strong Condensation at their least strong limit cardinal. (Strong condensation cannot hold at any cardinals past the first Ramsey cardinal.) Moreover by a theorem of Woodin, under $\text{AD}^+ + V = L(P(\mathbb R))$, for a Turing cone of reals $x$, $\text{HOD}_x$ satisfies Strong Condensation at its least strong limit cardinal. This heuristically argues that Strong Condensation should hold at the least strong limit cardinal in canonical inner models satisfying arbitrarily strong large cardinal axioms, assuming that such models exist. The reason is that the pattern observed in inner model theory to date suggests that these models should locally resemble the $\text{HOD}$s of determinacy models.

Here is the approximation to the existence of $h$ one gets assuming Strong Condensation at the least strong limit cardinal $\gamma$: There is a set $a\subseteq \omega_1$ and a function $g:\omega_1\to\omega_1$ such that for any $N\prec V_\gamma$ with $a\in N$, $g(N\cap \omega_1) > \text{ot}(N\cap \gamma)$. The rest of this answer consists of a proof of this fact.

Fix $f : \gamma\to H(\gamma)$ witnessing Strong Condensation at $\gamma$. The first step of the proof is cosmetic. One uses a theorem of Woodin which states that $f$ is definable over $H(\gamma)$ from the parameter $f \restriction \omega_1$. Let $a\subseteq \omega_1$ code $f\restriction \omega_1$. (It is an easy exercise to show that $f\restriction \omega_1\in H(\omega_2)$.) Then every $N\prec V_\gamma$ with $a\in N$ has the property that $N\cap H(\gamma)\prec (H(\gamma),f)$.

For $\alpha < \gamma$, let $P_\alpha = f[\alpha]$. Note that the $P_\alpha$ are increasing with union $H(\gamma)$, and if $M\prec (H(\gamma),f)$ and $\text{ot}(M\cap \gamma) = \alpha$, then the transitive collapse of $M$ is equal to $P_\alpha$. The structures $P_\alpha$ will play the role of the $L_\alpha$ hierarchy in Larson's proof.

For every $\xi < \omega_1$, let $g(\xi)$ be the least ordinal $\alpha$ such that there is a surjection from $\omega$ to $\xi$ in $P_\alpha$. Suppose $N\prec V_\gamma$ and $a\in N$. We will show that $g(N\cap \omega_1) > \text{ot}(N\cap \gamma)$. Let $M = N\cap H(\gamma)$, so $M\prec (H(\gamma),<)$. Clearly it suffices to show that $g(M\cap \omega_1) > \text{ot}(M\cap \gamma)$. Let $H_M$ be the transitive collapse of $M$. Then $M\cap \omega_1 = \omega_1^{H_M}$ and letting $\beta = \text{ot}(M\cap \gamma)$, $H_M = P_\beta$. Assume towards a contradiction that $\beta \geq g(\omega_1^{H_M})$. By the definition of $g$, there is a surjection from $\omega$ to $\omega_1^{H_M}$ in $P_\beta$. But since $P_\beta = H_M$, this contradicts that $\omega_1^{H_M}$ is uncountable in $H_M$.

Not sure if this is what you're interested in, but let me take up the question of what large cardinal axioms the existence of $h$ is known to be consistent with. The answer is: all large cardinal axioms that are known to have a canonical inner model. Beyond that, the consistency question remains open: indeed, it remains open whether large cardinal axioms beyond the reach of inner model theory could outright imply the existence of a precipitous ideal on $\omega_1$, and thereby refute the existence of $h$.

It seems plausible, however, that $h$ in fact exists in the canonical inner models yet to be discovered. This is because one can construct a function that is almost exactly like $h$ (see the fourth-to-last paragraph below) using a general condensation principle due to Woodin called Strong Condensation, which likely holds in all canonical inner models. If $\kappa$ is a cardinal, Strong Condensation at $\kappa$ states that there is a surjective function $f : \kappa\to H(\kappa)$ of such that for all $M\prec (H(\kappa),f)$, letting $(H_M,f_M)$ be the transitive collapse of $(M,f)$, $f_M = {f}\restriction H_M$.

All the known canonical inner models satisfy Strong Condensation at their least strong limit cardinal. (Strong condensation cannot hold at any cardinals past the first Ramsey cardinal.) Moreover by a theorem of Woodin, under $\text{AD}^+ + V = L(P(\mathbb R))$, for a Turing cone of reals $x$, $\text{HOD}_x$ satisfies Strong Condensation at its least strong limit cardinal. This heuristically argues that Strong Condensation should hold at the least strong limit cardinal in canonical inner models satisfying arbitrarily strong large cardinal axioms, assuming that such models exist. The reason is that the pattern observed in inner model theory to date suggests that these models should locally resemble the $\text{HOD}$s of determinacy models.

Here is the approximation to the existence of $h$ one gets assuming Strong Condensation at the least strong limit cardinal $\gamma$: There is a set $a\subseteq \omega_1$ and a function $g:\omega_1\to\omega_1$ such that for any $N\prec V_\gamma$ with $a\in N$, $g(N\cap \omega_1) > \text{ot}(N\cap \gamma)$. The rest of this answer consists of a proof of this fact.

Fix $f : \gamma\to H(\gamma)$ witnessing Strong Condensation at $\gamma$. The first step of the proof is cosmetic. One uses a theorem of Woodin which states that $f$ is definable over $H(\gamma)$ from the parameter $f \restriction \omega_1$. Let $a\subseteq \omega_1$ code $f\restriction \omega_1$. Then every $N\prec V_\gamma$ with $a\in N$ has the property that $N\cap H(\gamma)\prec (H(\gamma),f)$.

For $\alpha < \gamma$, let $P_\alpha = f[\alpha]$. Note that the $P_\alpha$ are increasing with union $H(\gamma)$, and if $M\prec (H(\gamma),f)$ and $\text{ot}(M\cap \gamma) = \alpha$, then the transitive collapse of $M$ is equal to $P_\alpha$. The structures $P_\alpha$ will serve as an approximation to the $L_\alpha$ hierarchy.

For every $\xi < \omega_1$, let $g(\xi)$ be the least ordinal $\alpha$ such that there is a surjection from $\omega$ to $\xi$ in $P_\alpha$. Suppose $N\prec V_\gamma$ and $a\in N$. We will show that $g(N\cap \omega_1) > \text{ot}(N\cap \gamma)$. Let $M = N\cap H(\gamma)$, so $M\prec (H(\gamma),<)$. Clearly it suffices to show that $g(M\cap \omega_1) > \text{ot}(M\cap \gamma)$. Let $H_M$ be the transitive collapse of $M$. Then $M\cap \omega_1 = \omega_1^{H_M}$ and letting $\beta = \text{ot}(M\cap \gamma)$, $H_M = P_\beta$. Assume towards a contradiction that $\beta \geq g(\omega_1^{H_M})$. By the definition of $g$, there is a surjection from $\omega$ to $\omega_1^{H_M}$ in $P_\beta$. But since $P_\beta = H_M$, this contradicts that $\omega_1^{H_M}$ is uncountable in $H_M$.

Not sure if this is what you're interested in, but let me take up the question of what large cardinal axioms the existence of $h$ is known to be consistent with. The answer is: all large cardinal axioms that are known to have a canonical inner model. Beyond that, the consistency question remains open: indeed, it remains open whether large cardinal axioms beyond the reach of inner model theory could outright imply the existence of a precipitous ideal on $\omega_1$, and thereby refute the existence of $h$.

It seems plausible, however, that $h$ in fact exists in the canonical inner models yet to be discovered. This is because one can construct a function that is almost exactly like $h$ (see the fourth-to-last paragraph below) using a general condensation principle due to Woodin called Strong Condensation, which likely holds in all canonical inner models. If $\kappa$ is a cardinal, Strong Condensation at $\kappa$ states that there is a surjective function $f : \kappa\to H(\kappa)$ of such that for all $M\prec (H(\kappa),f)$, letting $(H_M,f_M)$ be the transitive collapse of $(M,f)$, $f_M = {f}\restriction H_M$.

All the known canonical inner models satisfy Strong Condensation at their least strong limit cardinal. (Strong condensation cannot hold at any cardinals past the first Ramsey cardinal.) Moreover by a theorem of Woodin, under $\text{AD}^+ + V = L(P(\mathbb R))$, for a Turing cone of reals $x$, $\text{HOD}_x$ satisfies Strong Condensation at its least strong limit cardinal. This heuristically argues that Strong Condensation should hold at the least strong limit cardinal in canonical inner models satisfying arbitrarily strong large cardinal axioms, assuming that such models exist. The reason is that the pattern observed in inner model theory to date suggests that these models should locally resemble the $\text{HOD}$s of determinacy models.

Here is the approximation to the existence of $h$ one gets assuming Strong Condensation at the least strong limit cardinal $\gamma$: There is a set $a\subseteq \omega_1$ and a function $g:\omega_1\to\omega_1$ such that for any $N\prec V_\gamma$ with $a\in N$, $g(N\cap \omega_1) > \text{ot}(N\cap \gamma)$. The rest of this answer consists of a proof of this fact.

Fix $f : \gamma\to H(\gamma)$ witnessing Strong Condensation at $\gamma$. The first step of the proof is cosmetic. One uses a theorem of Woodin which states that $f$ is definable over $H(\gamma)$ from the parameter $f \restriction \omega_1$. Let $a\subseteq \omega_1$ code $f\restriction \omega_1$. (It is an easy exercise to show that $f\restriction \omega_1\in H(\omega_2)$.) Then every $N\prec V_\gamma$ with $a\in N$ has the property that $N\cap H(\gamma)\prec (H(\gamma),f)$.

For $\alpha < \gamma$, let $P_\alpha = f[\alpha]$. Note that the $P_\alpha$ are increasing with union $H(\gamma)$, and if $M\prec (H(\gamma),f)$ and $\text{ot}(M\cap \gamma) = \alpha$, then the transitive collapse of $M$ is equal to $P_\alpha$. The structures $P_\alpha$ will play the role of the $L_\alpha$ hierarchy in Larson's proof.

For every $\xi < \omega_1$, let $g(\xi)$ be the least ordinal $\alpha$ such that there is a surjection from $\omega$ to $\xi$ in $P_\alpha$. Suppose $N\prec V_\gamma$ and $a\in N$. We will show that $g(N\cap \omega_1) > \text{ot}(N\cap \gamma)$. Let $M = N\cap H(\gamma)$, so $M\prec (H(\gamma),<)$. Clearly it suffices to show that $g(M\cap \omega_1) > \text{ot}(M\cap \gamma)$. Let $H_M$ be the transitive collapse of $M$. Then $M\cap \omega_1 = \omega_1^{H_M}$ and letting $\beta = \text{ot}(M\cap \gamma)$, $H_M = P_\beta$. Assume towards a contradiction that $\beta \geq g(\omega_1^{H_M})$. By the definition of $g$, there is a surjection from $\omega$ to $\omega_1^{H_M}$ in $P_\beta$. But since $P_\beta = H_M$, this contradicts that $\omega_1^{H_M}$ is uncountable in $H_M$.