If you have an established average for each hand, you would declare before you started which hand would be used and use that average for determining which division you were in or how much handicap you had. You probably could not bowl, say, doubles and singles left-handed and teams right-handed and combine the two for all-events. You might be able to enter for each event twice, once with each hand, though the rules on how many times you can cash might hit you. Most tournaments have rules stating that you may only cash once in any event.
Basically, each hand is its own "person" and you'd have to decide which person was going to bowl that tournament.
I'm not sure what "has a 155 average left hand and can shoot low 700", that would indicate to me that the person has possibly sandbagged with the left hand. I don't know any legitimate 155-average bowlers who can walk in and expect to shoot a low 7.
A tournament might have a rule specifying that you use the highest average hand or that if you have both a left- and right-handed average, you must use your "dominant" hand.
SH
