Here is my attempt: Let us work over $\mathsf{ZFC^-} + V=L_{\omega_1}$, that is, $V=L$ holds and every set is hereditarily countable.
Then consider the following "minimal" definition of the $\diamondsuit$-sequence:

$(S_\alpha,C_\alpha)$ is the $<_L$-least pair $S_\alpha\subseteq\alpha$, $C_\alpha\subseteq \alpha$ such that $S_\alpha\cap\xi \neq S_\xi$ for all $\xi\in C_\alpha$. Otherwise, $S_\alpha=C_\alpha=\alpha$.

It gives a $\Sigma_1$-definable class $\langle S_\alpha\mid \alpha\in\mathrm{Ord}\rangle$. Let us call this sequence the *minimal $\diamondsuit$-sequence.*

I claim that the minimal $\diamondsuit$-sequence $\langle S_\alpha\mid \alpha\in\mathrm{Ord}\rangle$ really gives a diamond sequence in the following sense:

For every definable class $X=\{\alpha\in\mathrm{Ord}\mid \phi(\alpha,p)\}$ and a definable club $C=\{\alpha\in\mathrm{Ord}\mid \psi(\alpha,q)\}$, there is $\alpha\in C$ such that $X\cap\alpha=S_\alpha$, that is,
$$\exists \alpha [\psi(\alpha,p)\land S_\alpha=\{\xi<\alpha\mid \phi(\xi,q)\}.]$$

It mostly follows from the standard proof, which is available in, say, Jech's book. Assume the contrary that our "minimal" $\diamondsuit$-sequence is not a diamond sequence. Then we have formulas $\phi(\alpha,p)$ and $\psi(\alpha,q)$ such that $C=\{\alpha\mid \psi(\alpha,q)\}$ forms a class club and
$$\forall\alpha [\psi(\alpha,q)\to S_\alpha\neq \{\xi<\alpha\mid\phi(\xi,p)\}].$$

Now assume that both of $\phi$ and $\psi$ are at most $\Sigma_n$ for some $n\ge 1$. Let us observe that $\vDash_{\Sigma_n}$ enumerates all possible $\Sigma_n$-formulas, and is $\Sigma_n$-definable.
Thus we can "well-order" classes of the form $\{x\in L\mid \vDash_{\Sigma_n}\phi(x,p)\}$ in a manner that is coherent with the well-order over $L$.

I should specify the well-order over $L$: Let us recursively define $<_\alpha$ over $L_\alpha$ as follows. For technical convenience, let $\operatorname{Def}(X)$ be the set of all subsets of $X$ definable by formulas of the form $\exists x_0\forall x_1\cdots \mathsf{Q} x_{n-1}\phi(\vec{x},y,p)$ for some bounded formula $\phi$, where $\mathsf{Q}$ is an appropriate quantifier. Let us call such formulas *normalized $\Sigma_n$-formulas.*

Fix an enumeration $\langle\phi_\nu\mid \nu<\omega^2\rangle$ of all normalized formulas of the aforementioned form such that $\{\phi_{\omega\cdot n+k}\mid k<\omega\}$ is the set of all normalized $\Sigma_n$-formulas. Now consider the following order:

$<_\delta=\bigcup_{\alpha<\delta} <_\alpha$ if $\delta$ is a limit, and for $X,Y\in\operatorname{Def}(L_\alpha)$, we say $X<_{\alpha+1} Y$ if either

- $X,Y\in L_\alpha\land X<_\alpha Y$, or
- $X\in L_\alpha \land Y\notin L_\alpha$, or
- If $\mu, \nu<\omega^2$ are the least ordinals such that $X=\{x\in L_\alpha\mid \phi_\mu(x,p)\}$ and $Y=\{x\in L_\alpha\mid \phi_\mu(x,q)\}$ for some $<_\alpha$-minimal $p,q\in L_\alpha$, then either $\mu<\nu$ or ($\mu=\nu$ and $p<_\alpha q$).

With the help of $\vDash_{\Sigma_n}$, we can enumerate all $\Sigma_n$-definable classes, and we can well-order $\Sigma_n$-definable classes with respect to the above order. However, unlike the usual $<_L$, the well-order between classes is not $\Sigma_n$. I have not computed precisely, but I suspect its complexity is below $\Sigma_{n+3}$.

Thus we can pick the minimal $(\phi,p)$ and $(\psi,q)$ under the above well-order satisfying the failure of "the minimal $\diamondsuit$-sequence" being a diamond.

Now consider the countable $\Sigma_{n+99}$-Skolem hull $M$ of $L$ containing the transitive closure of $p$ and $q$. By Mostowski's collapsing lemma and condensation, we have the collapsing map $\pi\colon M\cong L_\delta$ for some ordinal $\delta$ such that $\pi(p)=p$ and $\pi(q)=q$.

Since our minimal $\diamondsuit$-sequence is $\Sigma_1$-definable without parameters, its definition is absolute between $L_\delta$ and $L$. Furthermore, since $\pi$ fixes $p$ and $q$, $C$ and $X$ are also absolute between $L_\delta$ and $L$.
In addition, $L_\delta$ thinks the following holds:

- $C\cap\delta=\{\xi<\delta\mid \psi(\xi,q)\}$ is a class club.
- For every $\xi<\delta$, if $\psi(\xi,p)$, then $S_\xi\neq \{\eta<\xi\mid \phi(\eta,p)\}$.

From the first fact that $C\cap \delta$ forms a class club, $\delta\in C$. Hence by the assumption, $S_\delta\neq \{\xi<\delta\mid \phi(\xi,p)\}$. However, the second fact with the minimality of $(\phi,p)$ and $(\psi,q)$ tells $(X\cap \delta, C\cap\delta)$ is the $<_L$-least pair witnessing the definition of the minimal $\diamondsuit$-sequence, so $X\cap \delta = S_\delta$, a contradiction.