FuncCPA is a recent security notion in which the CPA game is extended by a functional re-encryption oracle in order to model setups in which a server performing FHE computations is allowed to interactively delegate part of the computation back to the client. In this paper, we study funcCPA-style variants of several CCA security notions, including CCA1 and the more recent vCCA security. Contrary to the CPA case where a strict separation holds between CPA and funcCPA, we show that these new variants are equivalent to their respective originating CCA security notions. Interestingly, funcCPA-style security notions also model setups where, rather than delegating part of the encrypted domain computation all the way back to the client, the server has the ability to perform this delegation towards an honest or semi-honest client proxy it hosts, such as a secure enclave. We then provide a number of blueprints for achieving both FHE-like capabilities and advanced CCA security properties which may then meaningfully be implemented by leveraging on the combination of a partially homormophic scheme and a semi-honest non-colluding enclave hosted within the server performing the encrypted domain calculations.