Itma, your proposal is out of question. It does not work.
Sorry?
DLQ, squash her as always, and then provide test frame with eggs/open larvae? They
would make emergency QC(s). And, in time, sort themselves out.
That
would work.
And one could add the refinement of selecting a QC started on the youngest of larvae (culling the first-capped) QCs, but there should be no question that the colony
could be saved by this method.
Sure, a ready-made new Q would be better/quicker ...
But the only immediate source of a ready-made Q seems to be one of the splits.
Combining with that would be yet another option.
Not splitting the second hive, but instead combining with the (DLQ removed) drone-brood colony
would have been a better route to follow. And then, that stronger colony could have been split at some point in the (nearish) future. But that option is no longer possible.
Without seeing the situation in person, I can't say whether or not there would still be a possibility of combining all three of these poor colonies at this point.
There are several different plausible scenarios.
Without seeing the detail of the situation, I don't think anyone can say which would be best.
The important point is that "Shaking out" - as per the title of this thread - is neither a necessary nor a wise response to a Drone Laying Queen which is the situation described in the first post.
"Shaking Out" is what you do in response to a DIFFERENT situation, Laying Workers.
DLQ - remove and terminate her (she is no use any more), and then either requeen (by whatever method) or combine with a Queenright colony.
LW - shake out -
they are "doomed".
Both situations give rise to a lack of worker brood and just drone brood.
They can be distinguished by the laying pattern (or not) and the presence of a Q (or not).
Quite different situations, with quite different responses.