Thanks for your inputs. Looks like parking the car in the airport first to take care of my luggages and then returning the car would be best so far. I haven't asked the airline but I'm pretty sure they will have no problem receiving my additional luggages after check-in as long as I give the agent a heads-up while checking in.

Maybe I can try going into the return facility to see I can do the whole thing there and if not, go to the short-term parking lot?

Could anybody let me know if that would be possible, that is to go to the return lot and then come back out without actually returning the car if it becomes clear that I have to take multiple trips and they are not willing to hold my luggages?