You're going to open a can of worms here on UK rail prices, breaking journeys etc.

I've had a quick look and think you might find it's cheaper to go straight to Bristol & overnight, then swallow the cost of the second night in London.

Here's why...
You avoid having to take a peak time train to get to Bristol in time for lunch
You can buy a cheap advance one way Bristol-London
You're not having to go back out to Reading & in again (and many hotels will require a taxi from the station, too).

At a guess I think that will save you north of £100, which will cover the difference between a reasonable hotel in Reading and one in central London. Check out Grange Hotels - they have a bunch of converted townhouses near the British Museum that I was very pleasantly surprised with this time last year. They seem to be cheap outside the tourist peak.