You mention 'set-theoretic pluralism', so you already know that standard classical logic, with the standard interpretation via model theory already is such a logic.
Those things that are true in all models of your axioms are true, those things that are true in no model are false. If you can create both a consistent model that contains an assertion and one that does not, then it is, by definition, neither true nor false, meeting neither criterion.
This already establishes all the things you ask about, including the semantics, which are determined by that definition of true and false. It also kind of prescribes how the logic 'works' -- one must mark out and set aside independent hypotheses as potential seed axioms and not use facts within the same proof or construction that require contradictory axiom collections.
So I miss the point of the question. I assumed you expected some kind of finite or transfinite proof-procedure for such a logic, but your responses indicate otherwise. Perhaps it is hidden in the motivation.
In this approach to modern set theory, one need not be constrained to anything as weak as intuitionism or constructivism in deductions or in the 'meta-language'. You can escape it in many ways, but two are obvious.
First, this definition of truth is based upon the construction of internally consistent example universes, and not on deduction. So things like the law of the excluded middle can be taken as axiomatic, and included as part of the definition of consistent. You need only 1) believe that isomorphic models really act the same and 2) give up the notion that there is a single over-arching meta-model of the entire universe which is internally consistent.
Second, you can stretch the notion of construction to some degree. The most basic models, L and V, include ordinals within the models. This gives you transfinite induction and, therefore, transfinite proof theory, which allows for 'constructive' proofs about a wider range of things. Given that convention, you can presume a tower of 'large cardinal axioms' reaching up to Woodin's 'Ulitmate L' which increase the power of infinite proofs by using the idea that one of the 'union steps' in any transfinite deduction will happen over a witness to the presumed axiom.
Also, I am not claiming the logic of model theory is free of confusion, only that it does, in fact, exist. One bizarre aspect of the semantics here that you call out by labeling the two layers is that the model construction happens in one set theory while the models themselves represent instances of another.
For instance, "the axiom of determinacy of infinite games" contradicts the axiom of choice. Studying the axiom of determinacy, we can create a space of models of it. Then in all those models the axiom of choice is necessarily false. But we create them embedded in a world where we assume the axiom of choice is true, and the semantics of models' existence allow for it. The semantics say, then, that the embedded proofs require it to be false, but our knowledge of those proofs is contingent upon it being potentially true. We do so because the universe where it is false has less freedom, so we are entertaining a superset of the models that would matter if it happened to be false. If the extra one's aren't real, no loss of credibility ensues.
But what if we did the opposite? We would have truths about the axiom of choice knowable only subject to its falsehood. The semantics admit such a thing, but whether it has any real meaning is highly questionable.
So far, we have amazingly found that our identified independent axioms clearly have a 'bigger' and a 'smaller' side, or they form 'towers' of freedom, like the tower of large cardinal axioms, or the tower that has "finitism, determinacy, projective determinacy, hierarchical determinacy, hierarchical choice, ramified choice, choice" and clearly goes from smaller to larger worlds.
They somehow do not have confluence points where it becomes ambigous which version of the world would 'admit more models'. But surely that is simply the human lack of imagination at work? It seems unreasonably convenient.