Is that the reason? Or is it that permitting hotel courtesy buses, in addition to the many public buses that serve Heathrow, would cause significant congestion at the airport?

I agree with the suggestions that it's not necessary to stay at T3 for a flight departing T3. My preference is for the Premier Inn - or one of the other hotels - at T4, which has a quick and frequent Underground connection to T3. Or stay in central London an extra night and take the Underground, Elizabeth Line or Heathrow Express the following morning. Or use one of the many hotels just outside the airport, along Bath Road, if the OP isn't averse to taking a public bus from outside the hotel to the airport terminal.