I'd argue what you ultimately get is a handful of near axiom As and Bs that you can prove in the smaller logics and for any sufficiently interesting statement you end up in combined logics that lose all the nice properties you were hoping to have. The involved proofs won't be justifiable because they lose the properties that gave the justifiability that came from the smaller non-union logics.
It's not a promising approach if the only statements of the prover you can justify are the simplest and most basic ones.
I'd argue that you ought to be able to avoid working in powerful logics; instead you'd end up using A (or an easily-derived consequence of A) as an axiom in the proof of non-trivial statement B and such, while still keeping to some restricted logic for each individual proof. This is quite close to how humans do math in a practical sense.
But avoiding working in the powerful logics is akin to working in a single logic as much as possible without merging any. So you've lost the benefit of multiple logics that you're originally claiming and you're back in my "use one logic" case.
There are real difficulties here, and you're right to point them out. But I'd nonetheless posit that staying within a simple logic as much as possible, and only rarely resorting to "powerful" proof steps such as a switch to a different reasoning approach, is very different from what most current ATP systems do. (Though it's closer to how custom "tactics" might be used in ITP. Which intersects in interesting ways with the question of whether current ITP sketches are "intuitive" enough to humans.)
It's not a promising approach if the only statements of the prover you can justify are the simplest and most basic ones.