The answer to both my question is that "adding conductors does not change anything". Olivier has already discussed this for the Principal Ideal Theorem, and for Hilbert 94 this is proven by Suzuki in
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1118782786 (the link does not work)

The whole point is that Hilbert 94 follows from a generalization of Furtwängler theorem, saying that the kernel of the transfer map $$\mathrm{ver}:G^\text{ab}\to N^\text{ab}$$ has order divisible by $[G:N]$, and this for all normal subgroups $N$ containing the commutators $[G,G]$ (or, equivalently, such that the quotient by them is abelian). The case $N=[G,G]$ is Furtwangler's theorem and applying it in my setting with $N=\mathrm{Gal}(H_L(\mathfrak{F})/K)$ in the notations of the question (now $H_L(-)$ are ray class fields of $L$) shows that $\iota$ is never injective.

The answer to both my question is that "adding conductors does not change anything". Olivier has already discussed this for the Principal Ideal Theorem, and for Hilbert 94 this is proven by Suzuki in
https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-121/issue-none/A-generalization-of-Hilberts-theorem-94/nmj/1118782786.full

The whole point is that Hilbert 94 follows from a generalization of Furtwängler theorem, saying that the kernel of the transfer map $$\mathrm{ver}:G^\text{ab}\to N^\text{ab}$$ has order divisible by $[G:N]$, and this for all normal subgroups $N$ containing the commutators $[G,G]$ (or, equivalently, such that the quotient by them is abelian). The case $N=[G,G]$ is Furtwangler's theorem and applying it in my setting with $N=\mathrm{Gal}(H_L(\mathfrak{F})/K)$ in the notations of the question (now $H_L(-)$ are ray class fields of $L$) shows that $\iota$ is never injective.

The answer to both my question is that "adding conductors does not change anything". Olivier has already discussed this for the Principal Ideal Theorem, and for Hilbert 94 this is proven by Suzuki in
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1118782786

The whole point is that Hilbert 94 follows from a generalization of Furtwängler theorem, saying that the kernel of the transfer map $$\mathrm{ver}:G^\text{ab}\to N^\text{ab}$$ has order divisible by $[G:N]$, and this for all normal subgroups $N$ containing the commutators $[G,G]$ (or, equivalently, such that the quotient by them is abelian). The case $N=[G,G]$ is Furtwangler's theorem and applying it in my setting with $N=\mathrm{Gal}(H_L(\mathfrak{F})/K)$ in the notations of the question (now $H_L(-)$ are ray class fields of $L$) shows that $\iota$ is never invective.

The answer to both my question is that "adding conductors does not change anything". Olivier has already discussed this for the Principal Ideal Theorem, and for Hilbert 94 this is proven by Suzuki in
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1118782786 (the link does not work)

The whole point is that Hilbert 94 follows from a generalization of Furtwängler theorem, saying that the kernel of the transfer map $$\mathrm{ver}:G^\text{ab}\to N^\text{ab}$$ has order divisible by $[G:N]$, and this for all normal subgroups $N$ containing the commutators $[G,G]$ (or, equivalently, such that the quotient by them is abelian). The case $N=[G,G]$ is Furtwangler's theorem and applying it in my setting with $N=\mathrm{Gal}(H_L(\mathfrak{F})/K)$ in the notations of the question (now $H_L(-)$ are ray class fields of $L$) shows that $\iota$ is never injective.