"but we're about to head into a dearth, and the remaining
(all-adult) population, which is numerous right now, is about to
start aging out. "

In my experience queenless bees do not age very fast at all.  I had a queenless hive last six months last year.  I think the thing that finally killed that nuc was getting robbed out.  When bees are not raising brood they just do not seem to age at the normal rate.  I think your main concern is to keep them from going laying worker which in general happens after they have been broodless for about a month.  If they have laying workers introducing a queen is real hard because they behave like they already have a queen.  Giving a frame of eggs and open brood once a week and carefully removing all emergency queen cells can keep them from going laying worker forever.

I think it is also possible you will have a new queen laying in the next few days.  I think based on what you said that is has been 28 or 29 days since the split.  If so you still have a few days left to be positive the hive really is queenless.  Is it possible a new queen is laying right now and you missed seeing the eggs?  That is easy to do.  I have done it more than once.

Dick

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