trip planning questions - and if there is a resource I missed where I can find this info, feel free to point me in the right direction.

Booking for early 2023. Will arrive LHR late Tuesday evenng. Eventually ending up in birmingham the next day. We will need a car for the duration so...
Plan is to rent a car at LHR and then drive up to Birmingham on Wednesday.
Since we arrive late at night, I was thinking it might make more sense to stay at an airprot hotel and pick up the rental car in the am then head out. (First question: does that make sense?).
To do that, is there an on site airport hotel that might make more sense? Is closer to the rental agencies?
We are flying in on AA metal from ORD. Usually T3 but is that changing /will that imapct suggestions for hotel?

Is this logistically not optimal? Better ideas?

I looked at train to Birmingham adn renting a car there and that was cost prohibitive. Especially since the car wuold be returned to LHR

TIA