Buckingham Palace on Sunday

It will be tight but doable depending on flight arriving on time, no delay in getting to gate, how long in Border Control, etc. Do not drive into London! Even on a Sunday it’s likely to be busy.

Amend Option B and book a hotel near Buckingham Palace. Take Heathrow Express, drop bags at hotel, do Palace tour then head to hotel for dinner and sleep. Take the train to Bath the next morning.

You are seriously crazy to consider driving in London after a transatlantic flight, no rest or food, do a tour, then back to Heathrow and then drive to Bath on one of the busiest roads in the country.

You will also pay a congestion charge of 15 GBP for driving in central London.

Where do you plan to park?

Buckingham Palace is usually open to the public when the monarch is not in residence. That has changed with the death of the Queen.

Right now,, Buckingham Palace is closed as is the ticket office. Not sure if it will open for the week following the mourning period. I doubt it with Charles in residence.

Good point. This likely for naught. One would think the King will be there based on current events.

Buckingham Palace is closed until further notice. If you have tickets for the State Rooms, you should receive an email with details of a refund. Rct.uk

No email yet ... but like someone said, The King will likely be in residence even if it reopens in four weeks time when we are there

I think the advice to spend that night in London is good, and so is the advice not to drive (whether in London or to Bath) the day after an overnight flight. You could get a train to Bath the next morning and pick up the car there for whatever driving plans you have elsewhere. Or use the Heathrow Express back to the airport, stay in one of many hotels there, and get the car the next morning to drive to Bath.

Obviously Buckingham Palace's opening schedule has changed and I doubt if it will be open three weeks from tomorrow, but you might wait to hear about your reservation. I expect the new King will spend more time there than his mother did in her last years, but I hope there will be times when the public can tour the state rooms. That was a highlight of my last visit to London.

Checking the website, the palace is closed and there is no mention of when it will be open again. See here for all the details.

https://www.rct.uk/important-information

I think I would stick to your initial plan and head straight to Bath.

I would try to change your rental to be picked up in Bath, to avoid driving immediately and also to avoid an extra night in London. The extra hotel change really isn't necessary if you take a train to Bath as soon as you arrive.

You're already jetlagged, so continuing to travel that day will still be productive and then you can start fresh in Bath the next morning!

Enjoy your trip!

PQ

After spending time in Windsor, right near Heathrow, it baffles me why Rick's suggested going to Bath right away. With so many easy bus and taxi options to Windsor, baffles me why he suggests renting a car in Bath.
Heathrow Express will get you to Paddington, and there are lockers there and a load of nice little hotels nearby, kind of tricky to find where the taxis park (have to take an elevator/lift up to a different area), but asking for directions will help. Found the bus connections not the very best, one of the stops is at the whole other end of the entire station and it doesn't tell you that on the bus schedule, so probably taxi is best if you're in a time crunch and overtired.
Whether you should or shouldn't go to Buckingham Palace is up to you, really. People often don't listen to their own inner voice that says what you really want to see and do, then they make up this frantic tour of what others tell them they SHOULD want to see. When you pick your very favorites, for whatever reasons that YOU have (not other people), just schedule them geographically close together, make sure you are mapping things out. Just getting out of Heathrow's a BIG project, none of these distances you describe are to be underestimated.
If you adore the idea of going to Bath, then you'll get to see BP from the outside, that could be OK. If you love the inside of BP, then hop on the Heathrow express and find a friendly taxi, maybe just do your London days first and go to Bath later. The lowest-stress option wins. :)
Have FUN.

Thanks to everyone for the gift of your experience. I think we've decided to forego the BP Tour in the interest of sanity and head to bath.