4th vote for going directly to Bath!
I love Bath and although many seem to "do" Bath as a day trip from London, there is much more to it than that, plus you only have a full day in Bath on the Best of England tour.
I see the Holburne Museum is having an exhibition feature work by Rodin and Degas if you are interested.
If you all are Jane Austen fans the JA Festival is from Sept 9-1. That's kind of fun because people wander around town in their Regency dress! There are some special walks and exhibitions associated with the festival.
https://janeausten.co.uk/pages/festival
(PS The Jane Austen Centre is NOT worth seeing - she didn't live there and they have nothing original of hers so for me it was a complete waste of time. I DID do one of their sponsored walks during the Festival several years ago and it was very good!)
If you are a garden person or enjoy architectural features, Prior Park is neat although it is an uphill walk to the entrance gate,lol. Fortunately then you can walk downhill thru the gardens!
Here's the website for the TI office.
https://visitbath.co.uk/
The Best of England does not go to Stonehenge so visiting there is an option and there are several ways to do it. There are day trips with MadMax, a local company using small vans for day trips. It's also really easy to take the train to Salisbury, pick up the shuttle out to the stones from the train station and stop by Salisbury Cathedral after Stonehenge and before you catch the hourly train back to Bath.
AND there is literally nothing at the airport, well except the airport and a bunch of hotels. IF you decide on this course, I'd stay in Windsor - take a cab there and back to the airport to catch the National Express coach.
BTW, long distance buses in England are nothing like the dismal Greyhound buses here! They are comfortable, safe, clean.