fonzie,
this is quite explicit in the rules. The basic intepretation is that if a bowler uses his other hand, it is equivalent to a new person bowling. He needs a new sanction card, indicating which "other" hand he used. It won't and CANNOT count towards his present average.
Since this is explicit in the ABC rules, you cannot, as far as I know, pass a league rule allowing his bowling with the other hand to count towards his current average. You can modify rules, you can pass rules not in the ABC rule book, but you cannot go completely against the ABC rules, by allowing something that the ABC forbids. NOTE that it does not forbid his using another hand; it just states how this will be used.
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"We get old too fast, and too late, smart."