All the 747's at BA have now had the AVOD update so you will have the AVOD system on alright...now as to whether or not it will work on the day is entirely in the hands of the gods I am afraid!!! LOL!
Your flight is currently showing over 50 seats available also so you may have plenty of room. However it is still a fortnight before your flight so that could change drastically in that time. Best bet is to get online EXACTLY 24 hours before departure to secure yourself a decent seat.
LAX seems to be awful all the time in my recent experience and the upgrading of the immigration desks and baggage belts is really not helping the whole process (or evening brightening it up...even the new bits look dreary still)!!! To be honest I am not sure how it would be at lunch time on a saturday but just make sure you make your way down ASAP from the aircraft!