Classes with oracles forcing the caller to immediately accept (or reject)?
The question is not entirely clear to me. However, concerning the example that is spelled out more precisely: if a language is recognizable by a poly-time machine with a SAT oracle which must accept whenever the oracle answers “yes”, it is in fact in NP. First, observe that regardless of the oracle answers, we can simulate in polynomial time the only possible run of the algorithm by pretending all answers are “no”. (In particular, we can compute the list of all oracle queries beforehand.) Then, the original algorithm accepts iff one of the oracle answers in the simulated run is actually “yes”, or the simulated algorithm accepts in the end. This is an NP property.

More generally, if we use an arbitrary oracle $A$ in the above scenario in place of SAT, a language can be computed in the indicated way iff it is poly-time dtt-reducible to $A$.