I wouldn't. Formic acid can damage brood. Also queen rearing is warm weather activity, and formic pro has a temperature range limitation (54-84°F). You know how our springs are here, we alternate hot and cold, sunshine and snow/frost, sometimes on the same day, until the end of May, then a week later it is a scorched earth Summer.
I'm planning on doing some splits putting the old queen and nurse bees in new hive and foragers and brood in old hive. The new hive wouldn't have capped brood yet so it would be a good time to treat to get them all quickly. The old hive wouldn't be laying and trying to make a queen so if it didn't hurt the forming queen, it also would be a good time to kill mites. I would start around mid april and I don't think it would get too hot in the next two weeks.
first, you want nurse bees to be making queen cells. they are the ones that produce royal jelly best. adjust to leave old queen with foragers. advise to leave one frame of open brood with them to hold them in the box otherwise you risk abscond. search taranov split or fly-back split.
your attempt to hit the mites when there is no brood would be better suited to use OAV or OAD.
if you are going to use formic i would treat before splitting. the higher population of bees will be able to move the vapors better. with formic the caps will be penetrated to kill mites in the brood. there is risk of queen loss if formic is not used properly, mostly temp related. so check for a laying queen 7-10 days after applying treatment. if you find queen cells then make splits with them. if you do still in fact have a laying queen then commence with your above plan using mite-free bees!
I would start around mid april and I don't think it would get too hot in the next two weeks.
Got it. I thought you were going to use it in the hive that was raising a new queen. I second what Coal Reaper suggested, treat before hand. Last year we had a late spring and didn't see drones here along the Wasatch front until mid May. This year spring looks to be about three or four weeks ahead of where we were last year. Make sure you are seeing drones before you make the split.