Possible, but quality may vary
I actually am aiding a local small private high school in attempting this. However, we were forced more by drought and a lack of funds to maintain a cool season field in this area. We started the beginning of this summer, so we weren't able to spray out the cool season. Instead we just shifted practices to encourage the bermuda. Right now, it's still pretty rough. Spraying out the cool season grass would have helped a lot.
I'm not 100% familiar with your climate there, but it sounds somewhat similar to what I deal with in southern Kansas. We can grow everything, but can't grow anything. Meaning: warm & cool grow here, but struggle equally in the extremes.
These decisions are always so dependant on the millions of variables your facing, but I would encourage the switch over to bermuda. I would spray it with glyphosate in March or April(whenever things are alive, but the bermuda is still dormant). Once the bermuda starts greening up I would start to hit it with a steady source of nitrogen through the summer. Keep it mowed at 1" or a little above. The main key is to keep encouraging lateral growth.
My only concern would be that sometimes native bermuda doesn't respond to traffic as well as some of the turf types. However, if you have a very limited budget and you're having issues with the cool season being extremely patchy bermuda would be a good way to go.
But as to your original question of "How it would turn out?"; unless you have bermuda almost all over the field right now it might be pretty rough the following year unless you come in with some sprigs or sod to help the field out. It's certainly feasible though.