Commuting to London from Bath is doable - it's tiring but ok. I've done it. Most people who commute to London live in 'desirable' Widcombe or Bathwick areas (you will pay a premium in both), because these are a 10 min or so walk to the station. The schools are full of ex-Londoners and commuting parents. Paying to park at the station will be expensive, and there often won't be spaces, and there's no on road parking at all.
The issue will be getting the children to Kingswood. You really want to be on the same side of Bath - so school, house and station on the south side. We have to get one of ours across town and it's a nightmare - been known to take 45 mins from the London Road to station area at rush hour. The London Road/Firestation area is often solid. A cycle route along the canal is possible but not great every day in bad weather, and you wouldn't want to go up the steep hills to Kingswood, Monkton or Prior Park at the end of that journey!
Private schools on southside are Monkton (that's a 15 min or so drive up a steep hill from the station), Prior Park/Paragon, which are 10 min walk - 10 min drive. Or King Edwards, in Bathwick.
Kingswood (and Royal High, for girls) are on the north side of Bath, again up a steep hill in the opposite direction.
Quite a few people use an electric bike due to the hills.
Someone mentions Camden as being slightly cheaper than Lansdown, and on the north side. The issue here is a general shortage of family houses - lots of 2 bed apartments, but you have to go quite far into Camden, on the the border with Larkhall to find family houses (mainly 3-4 bed small terraces, on steep hills).
Bathampton issue would be the traffic.