Bus or train from Bath?
Probably best to use the bus as if using the train, you would have to change to a bus in any case at Bristol Temple Meads as the airport does not have a rail connection.https://www.airdecker.com
In our opinion, neither is a good way just because of the hassle factor. It's really not too long a distance to rule out a private car service. It will cost a little more, but we think it is well worth it.
We used a private car service based in Bath to go from there to the Bristol airport (and it's much closer than Heathrow) and were very pleased. I don't remember the cost, but it was reasonable. The service we used was Chauffeur Travel, and the web site is https://chauffeurtravel.co.uk/
Send them an email explaining your needs and ask for a quote. Can't hurt to ask.
I would also recommend a taxi. There is a bus service but it will take a lot longer it’s not that frequent or cheap. The train station in Bristol is nowhere near the airport. Bristol airport is very poorly connected by any transport method.
We just took an Uber. It was £54 when we did this last year. I just like to make it “easy” for myself.