The hand to set your average with is the hand you must bowl with. You aren't normally allowed to switch without a compelling reason. However, you may bowl a different league with the other hand, setting a separate average just for that league.
You cannot, for instance, set an average with your left hand, then switch to your right hand (which you have a much higher average with) for "critical" games to take advantage of the handicap.
There is no rule, however, that prevents you from using whatever hand you like for the duration of the season. Your scores are valid, though there should be a mention in the record books that the average is for a particular hand. Your secretary is wrong. You started the league with that hand and you set average with it. You have only used that hand for this league.
You could bowl Tuesdays left-handed and Wednesdays right handed if you like. You would have a book average for each hand, and for tournaments where you must submit an average, you must specify which hand you will use.
SH