On one ticket your bags should be through-checked. I regularly do BA to LHR and then AA onwards and have never had a problem.
The Sofitel is a nice hotel and very convenient to T5 but definitely the most expensive at LHR. If money is no object go with the Sofitel and take the Heathrow Express to T3 the next morning. The other choice at the airport is the HGI at T3, also expensive and that would involve taking the train to T3 after you arrive and then walking to the hotel.
All other on-airport hotels are at T4 (Hilton, Crowne Plaza, etc.) but that involves trains aplenty.
Another option and the one I use most for connections are the hotels to the north of the airport in the free transport zone (Marriott, Hyatt Place, Holiday Inn etc.). After arriving at T5 you take the free 423 bus to any of those and the next morning take another free bus (the 105, 111, etc) to T3. See
https://www.businesstraveller.com/fe...e-travel-zone/ for the map with bus routes and hotels.
Finally, there are other off airport hotels (like the Hilton T5) connect to Heathrow by the Hoppa buses (9 Pounds return; kids are free) which also serve the hotels to the north of the airport if you don't want to catch the free buses.