Hello,
I can't help you with your second question, but regarding the first I'd say it's very possible to live in Bristol and come to Bath, but it would take dedication. The train takes about 20 minutes and come every 15-30 minutes. It is quite expensive (~£10) though, so you would want some sort of year pass for that. From the train station you would have to take a bus, with the U1 and U18 being the frequent services, up the hill to the university. The bus stop is right outside the station, although these buses can get very crowded in the morning, and so getting on at a mid-run bus station such as the train station may be difficult in the mornings at around 8-9am. I'd estimate that the bus takes a further 15 minutes but it stops right on campus.
You are right about Bristol being a larger city with more things to do. Bath has many nice parks for the kids and is very touristic, with a number of things to do. However, many of these activities would be for older people - I don't think your kids are at the age that they'd appreciate the Roman Baths etc. If anything, it is a personal choice that you have to make. You mention travelling because there would be more to do for the kids in Bristol, but what about living in Bath, with a quick commute to Bristol every so often for activities with the kids? Just a suggestion.
Whatever you choose, enjoy Bath, it's a great University and great place, as is Bristol. All the best.