Is not allowing you to stay overnight specific to T1 at LAX? IIRC LAX is "open 24hrs" so I don't think they would force you to leave. I've had previous flights that were significantly delayed and was rebooked on 6am departures (eg at DCA) and I simply found a quiet-ish part of the terminal to relax/sleep until the morning. Did something similar at PDX when I knew I'd be out late the night before and 6am flight; just went to the airport, through security, and waited.

I think the car option might end up being the best regardless... And luckily we both have TSA-pre so security shouldn't be too much of an issue. Thanks!