You won’t be allowed through flight connections in T3 with a bp for the next day - it needs to be a same day bp. You will therefore have to exit landside at T3 at that point, or just exit landside on arrival at T5.

T3 (like all other LHR terminals) is not open airside overnight. The exception is that some emergency provision is made for those small numbers who cannot go through immigration but have a next day connection, there is a small area set aside on T3 for that. But that’s not your situation.