I agree with perhaps getting a hotel in Paddington - you could get there on the Heathrow Express or even on the (cheaper) Elizabeth line in 20-30 minutes from T5, and 15-20 minutes back in the morning to T3. Range of hotels near Paddington for all budgets. Mercure comes up a lot.

You could be at your hotel by 8 or 830 and at least see something of London for the evening or eat somewhere a bit more interesting than an airport hotel bar. For a 930am flight you don't need to be back on the train until probably 7am the next day.

Definitely do-able. Would in any get a hotel overnight somewhere; hanging around landside overnight at Heathrow will not be comfortable or fun.