Would K400 really be enough to buy you something nice in Bath? I did a property search for Bath the other day, and the nice houses/areas all seemed to be K750.
Realistically, K400 in London is surely going to be outer London?
"Living" in a place comes down to the experience of living there, if you see what I mean.
K400 in London will probably buy you the experience of living close to all those central London amenities, and the reality of living in a suburb - which is not the same as living in central London, and that has its good and bad points.
Not living in central Bath - I have no real idea what that would be like. My mil lives in central Bath, and it is pretty wonderful: popping out to Waitrose of a morning, beautiful architecture, a cultural life on your doorstep (and, no, it isn't the cultural life of the capital, but on the + side, it's all on her doorstep), and gorgeous countryside, minutes away.
Living outside the centre of London has upsides - there is a vibrant life a short commute away; there is often a wide mix of people which is invigorating; transport is pretty good, so you can travel and widen your day to day experience further (on the other hand, you will have to travel, probably losing three hours of your day, every working day, so that may get old quite fast!!).
Bath, apparently, has a huge issue with secondary schools. It does have a problem with drug addiction - though that doesn't seem to have made a deep mark on the general culture of the city, so it may not be a realistic issue for you, at all.
Conversely, depending on what you want from your "London" experience, you can find pretty good schools, and peaceful little boroughs, or vibrant ones - London is very much a number of little areas, of very distinct character.
How lucky to be in a position to choose! I think you should think about what it is you would like to experience from the place you live, and have a great time looking. it must be lovely to not be tied to an area through work.