Okay, tried that: non-PoE switch at far-end does indeed then enable use of the passive PoE injector for the AP. So we have a decent fall-back now.

Removed that extra switch again, in favour of a direct PoE from the smart switch, and have now added both in-line repeaters with about 400' of cat5e after the second one. All still working nicely here.

So then.. various permutations of long cable runs, including about 400' of cable from the PoE switch to the first repeater, a short chunk from there to the second, and then a long-ish run to the AP via my in-wall wiring and several patch cables etc..

All still good.

Next up: I'll patch some of the in-wall segments together into a longer loop, and put that between the two repeaters, to better simulate the final situation.

EDIT: Done. All still working with a single repeater, as well as with two repeaters.