I don't know the LA area that well, but I can share that my DS once took a nap sitting on my lap on the trolley in San Francisco.
So if you can get to one of the "lines" (they're color coded) in LA, maybe you could just ride for a while in the afternoon?
Sorry, I live in Orange County, but I really know virtually NOTHING about public transportation in LA.
I agree that a 7 p.m. check-out is going to be a no-go, but you should be able to stow your luggage at the hotel.
DS '04 "Boogaboo"
DD '08 "Lilybear"