The delay is less than 4 hours, so no EU compensation is due, other than due care. However, the itinerary is on AA stock (given that it is an AA reward), not BA stock. BA just performed one flight out of the whole itinerary and does not subsequently become responsible for the entire itinerary. I do not see how BA can be held accountable for the problem of not getting you to LAX. Basically your complaint should be directed at AA.

As far as the unfortunate situation with the dogs is concerned, I always used to make sure I did not pick them up until the day after I arrived home for exactly this reason.