The sole question is whether AA properly reissued the ticket for the UA flight. No cash really changes hands between AA & US. But, you do need a valid ticket. This was easy once upon a time because you had a paper coupon on which AA wrote "Rule 240 to UA" and you schlepped over to UA with it and UA gave you a BP.
Rather than trying to play detective in a space where you will never get it figured out:
1. Claim for the hotel to AA. Either way, they cancelled due to MX and you didn't get out of LAX.
2. Make an IDB claim to UA (and do a webform compalitn to DOT). If you were properly ticketed and with a reservation and at the gate by T-15, but were denied, UA owes you for that. DOT will require UA to document the situation and, if AA is telling the truth, UA will have a valid ticket. If AA did not properly reissue the ticket to UA, there is no IDB on UA and no violation by AA (but nonetheless worth reporting to DOT as this is a simple and routine transaction).
While it's a PITA these days and busy agents hate it, one way to avoid this because it does happen is to ask the AA agent doing the rebooking to issue a paper coupon. You then schlep over to UA with it as in days of yore.