Good. That doesn't quite answer for the OP's case of merely not adding reals, though.

@MonroeEskew Good question.

Yes, probably best.

Cody, is your question a duplicate of that question, and if not, what is distinguishing for you?

See also: mathoverflow.net/q/281894/1946

Unfortunately, the acronym ZFA is ambiguous, since some people mean set theory with atoms i.e. urelements, and some people mean set theory with antifoundation AFA. In the atomic case, it is also ambiguous about whether collection+separation is included, or merely replacement, and whether there is a predicate for being an atom or whether they are indiscernible with the empty set.

Another keyword is "coalitions".

No, because all sets of reals show up in $V_{\omega+2}$. The $L$ hierarchy is much slower than the $V$ hierarchy.

It seems delicate to interpret ZFC in your theory, since the interpreted model will have Dedekind infinite sets, but to prevent those sets from giving rise to actual Dedekind infinite sets will mean that the interpretation must be using an equivalence relation (i.e. interpreting = nontrivially).

Interpreting ZFC is equivalent to interpreting ZFC+V=L, since L is definable in ZFC and so every interpreted model of ZFC also interprets a model of ZFC+V=L. Your theory doesn't seem to have its own L (unless this is really what you are asking?), so I think the L angle of your second question is irrelevant.

But then you wouldn't be constructing $L$, but interpreting a model.

What do you mean by constructing $L$ if you don't have $\omega$?