Last night in UK.

If it was me, I would probably stay in Bath but do all my packing the night before and get a really early start -- like be on the road by 6 AM. That would allow for traffic delays, potential wrong turn(s) finding the rental car dropoff, and lots of people ahead of you at security check-in.

If getting up that early doesn't work for you, turn in your car the night before and sleep in a hotel right by the airport, one that has a free airport shuttle. See https://www.heathrow-airport-guide.co.uk/hotels.html

Bath to Heathrow is likely to be about a 3 hour drive or longer if you are travelling Monday to Friday at that time of day ie rush hour. You will need a few minutes to turn your car in and you need to be in the airport for 9.35 latest. You would need to leave at 6am latest.

Personally, I wouldn’t want the stress of such a long drive on the busiest roads in Britain when I knew I had a flight to catch.

With returning the rental car and driving to LHR, you might want to consider staying in an airport hotel. It is suggested to be at the airport 3 hours ahead for international departure, this would make a very early departure from Bath. My opinion only, but I just wouldn't want the added stress. If you don't want to stay at an airport hotel, consider turning in your rental and booking a London hotel for your last night.

The Premier Inn near T4 is very reasonably priced and relatively new. You can walk to the terminal through an enclosed walkway in 15 minutes.

I wouldn't even think of renting a car for that trip. The Bath bus station is right by the train station, and has a very comfortable bus that goes directly to Heathrow. Several buses a day. If you go the day before your flight, stay at one of the hotels connected to Heathrow, or near your Heathrow terminal.

We took the Rick Steves Best of England tour in April. It begins in Bath. We arrived at Heathrow, walked through the terminal to the bus station, took the bus to Bath. Easy, fast, inexpensive.

Are you traveling with a tour group, or on your own? I know the Rick Steves Southern England tour ends in Bath. Are you finishing up that tour?

If you are traveling independently, I would consider making it one night in Bath, then traveling to Windsor, staying 2 nights. It is a very short distance to Heathrow from there. Day of your flight taxi to Heathrow; Price 22 pounds.

We stayed in Windsor for two nights before we departed. Took a taxi to the airport. This is how I know this would all work for you; we just did it.

I'd strongly suggest staying in London. Don't add any stress or possible monkey wrenches on departure day.

I agree with Rebecca about goi g to Windsor and staying there. So easy to get to the airport from Windsor.

When I’m in that part of the UK on business, rather than making that 2.5 hour drive in the morning, I drive to London, turn in the car, and spend the night before the flight in the city taking in some sights or just walking around. That way I can sleep in a little later, have a relaxed breakfast, then catch the tube to the airport. I stay near the Gloucester tube stop since it’s a direct link to the airport.

I like to leave from Windsor (for sightseeing too) or Slough/Hounslow (cheap hotel), very nearby towns.
Travelodge.co.uk has good priced hotels in both Windsor and Slough at least. Lots of options. I personally would spend 2 nights in Windsor since there's so much to see and do there. But if you really love Bath, and there's something important to do that last night there, and you're an early morning person, you could probably find a way to get to Heathrow in the morning. I'm not a morning person :( :) so I would be a failure at that.
Have a great trip.
-Alison