Let $\sf T$ be a theory that has as axioms every axiom of $\sf ZFC$, and every theorem of $\sf ZFC + [V=L]$ that is neither provable nor disprovable by $\sf ZFC$, whose addition or addition of its negation doesn't increase  consistency strength over $\sf ZFC$, and that is consistent with $\sf ZFC + [V \neq L]$.

> Are there models of $\sf T$ that satisfy existence of a measurable cardinal?

> Given that this theory imports many decisions made by the Constructibility axiom, is there a limit on large cardinal properties that can be satisfied by its models?