Well, I'm just brainstorming here but I can just think of two ways to go:
1)Open the bottle and wait until you think it has gone flat. Siphon to the 12 0z bottle and add a priming tablet or a carefully measured amount of priming sugar.
2)Get the 22oz beer and the empty 12 oz bottles as cold as you can without freezing and then transfer and cap as quickly and as carefully as you can.
I have personally been in your exact situation with the same question but I decided I didn't want to risk oxidation or anything else that could go wrong.
But there's nothing wrong with giving it a try. Just transfer an extra bottle or two and when the contest deadline approaches, taste it. If you still like what you have, then enter your beer in the contest.
Good luck!
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