My feet are very particular
Plan A: If it's cold and wet you'd want proper weather protection. Running shoes won't provide that per se, but do also know that modern boots don't need breaking in. Is a myth from the days of leather boots. I have, for example, taken a pair of Salomon S/Lab alpine-2 straight out of the box on the parking lot and done 4000m of altitude in 24 hours with no blisters.
Plan B: Ergonomics beats everything else. If you can't find a price/fit that is without compromise, I would suggest that you stick with what works for your feet and simply acknowledge the limitations. My background for this advice is that I've run mountain routes in Salomon Cross Speed 4 & 5 Gtx which include climbing grade 5.2 sections. Be quick to drop to all four if the surface is slippery to mitigate the risk of slipping/falling. If there is snow/ice left on the summit, compensate for the lack of grip with a pair of spikes for running shoes. The spikes weigh very little and and increases you safety margin a lot - particularly in combination with a pair of walking sticks with a carbonate tip and snow baskets.
If you choose B you'll have to manage you shoes more actively than others who have the extra protection that a boot gives if the weather is moist, as soaked footwear will suck energy out of you. Your feet need warm blood for the muscles and the only heat source to compensate for the energy loss will be you.
I'm very fit, and and can move fast to keep warm as I prefer traveling solo. If you're in a group that moves like a slug you probably won't have that option.
I hope this helps?
