Yes. Double brood it with the queen in the deep brood and the shallow brood above a Q/E. Check for queen cells and remove them (in the shallow) until no more can be produced. It may now be slow, dependent on flows, but it will be on double brood with a honey super in about three weeks (a bit more if there is any drone eggs in the shallow).
Normal summer weather might mean no warmth problems, but this year....it is however quite a strong colony on a brood and a half, so not as bad a s dumping a 5 frame nuc straight into a full deep with foundation.
I would likely place the brood centrally into both boxes and allow them to expand sideways onto the foundation, and if really cold put the new box below the current deep.
RAB