I live in Bristol (Redland/Cotham area, very near Cotham school) but visit Bath frequently. I would say Bristol is far better for older kids - Bath has a great central park, some fun museums, and the children's shows at the Egg are brilliant, but Bristol, being bigger, has much more going on, particularly for older kids (my DCs are 11 and 6, by the way). Bath does seem full of retired people and tourists a lot of the time (sorry - I fully expect anyone who lives there to correct me on this...).
Though having said that, I grew up on the outskirts of Bristol, which was not so great - not much public transport, so I hated always being dependent on parents for lifts. But if you are planning to live within the Redland Green/Cotham catchment areas, that should mean being within walking distance of much of central Bristol, or being on one of the better, more frequent bus routes, so teens and pre-teens can be fairly independent (my 11-yr-old DS goes a lot of places by himself now).
Catchment areas are tricky, and keep moving - don't believe the council website on the Redland Green catchment area: it is so oversubscribed that I think you actually have to live within about 500m to stand a good chance of a place. House prices tend to reflect that...
The Cotham catchment is much broader, and has some parts that are much cheaper than Clifton, eg down the hill from the school, towards Gloucester Road, which is lively and much more down-to-earth than Clifton. Also worth looking down the hill from Clifton in Cliftonwood - lower prices, very strong community spirit, and everyone I know there who applied to Cotham got a place this year. It's also handy for the Cathedral school, a private-turned-state academy which runs admissions on a kind of lottery basis.
My area can tick most of your boxes - community feel (if you avoid the roads with too many students), near green space, and plenty of period houses, though you may have to scale down your expectations on garden size....