Simulating DFA operations on non-regular languages with finite final state set
Your question is missing quite a few definition, but here is my interpretation.


  A language $L \subseteq \Sigma^*$ is finite-regular if there exists $C$ such that for all $n$, $L \cap \Sigma^n$ is accepted by some DFA with at most $C$ final states.


Note that if the minimal DFA for a language has $m$ accepting states, then every DFA for the language has at least $m$ accepting states (this follows from Myhill–Nerode theory). Therefore we obtain an equivalent definition if we replace "some DFA" with "the minimal DFA".

Using the product construction, you can easily show that the class of finite-regular languages is closed under union and intersection, and more generally under monotone set operations.