I have to agree with the majority of people in this thread. If you were to do that, with the difference in average, you might as well be two different bowlers bowling subbing under the same name. In any league that I am in, after you have subbed the first week under your most recent book average, you establish an average in that league. The next time you sub, you have to bowl under that average that you established, not your most recent book average. I can't imagine that any league secretary would let you carry two established averages, one right handed, one left handed, or let you sub under your most recent book average week after week even though you have already bowled a decent amount of games in the league. Imagine you bowl lefty the first time you sub, establish a lower average, and then decide to bowl right handed to take advantage of it. Then bowl lefty again to bring it a bit lower again. It would seem to be a way to perform average management without it looking too suspicious to others.
Even if you had pretty much the same average for both hands, I'd still be against it. Let's say you bowl in a 2nd shift league, you could watch the bowlers on the 1st shift, see how many rightys/leftys are on the pair and watch where they are playing. Then, you determine which hand you want to use based on whichever hand gets you closest to your most comfortable shot to the pocket.
As said earlier, if you want to bowl with both hands, bowl righty in one league and lefty in another. Unless you're injured, I see no legitimate reason to make the switch in the same league, and especially doing it on a week to week basis.
Current Average - 225