Let me not answer the question asked but add an important angle.

An L can be built already in ATR_0, which is the weakest theory that can do it convincingly (some coding involved).

You can't guarantee Powerset in that L, but it can well happen that this L acquires lots of uncountable cardinals. (All ordinals were "countable" in the initial model of arithmetic, but after the extraction of L, many original bijections between ordinals and N were left outside.)

I guess the best source for this is Simpson's "Subsystems of Second Order Arithmetic", parts VII.3 and VII.4.

Perhaps what is also very relevant to your thoughts is something called "the Feferman-Leví model" discussed on page 295 of Simpson's book and elsewhere.

Let me not answer the question asked but add an important angle.

An $L$ can be built already in $\mathsf{ATR}_0$, which is the weakest theory that can do it convincingly (some coding involved).

You can't guarantee Powerset in that $L$, but it can well happen that this $L$ acquires lots of uncountable cardinals. (All ordinals were "countable" in the initial model of arithmetic, but after the extraction of $L$, many original bijections between ordinals and $\mathbb N$ were left outside.)

I guess the best source for this is Simpson's "Subsystems of Second Order Arithmetic", parts VII.3 and VII.4.

Perhaps what is also very relevant to your thoughts is something called "the Feferman-Leví model" discussed on page 295 of Simpson's book and elsewhere.