I think that because there is not prizes given to actual accomplishments, you should be ok in having your cousin bowl.  It is just like all those disney ball leagues or sponge bob ball leagues, where all you do is participate and you get the same thing whether you finish first or last.  I used to manage a center and those leagues were filled with kids so I doubt that the proper information was given to USBC.  That was 3 years ago, but I doubt it has changed.  Keep bumping this post and I am sure that you will get a more official answer soon.
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I am not a pro-bowler, but I do play one on BallReviews.com

Yeah. Looks like based on the FAQ of the rulebook:

"175. Our center is starting a league that will give NFL jackets to all members who remain the full 30
weeks. The jackets will cost over $100. Can my children join with me without violating their USBC
Youth membership eligibility? Yes, since receiving the jacket has nothing to do with bowling skill, score or
position standing. Actually, the league members are paying toward the cost of the jacket in league fees each week.
The same holds true for ball, bike or merchandise leagues as long as they follow Rule 400 and every bowler
receives the same item."

Thanks for the insight guys. Didn't even bother to read the book ;P

I still would double check with usbc because in the last month or so there has been revision to rule 400. I believe something along the line a youth member cannot receive anything over the value of $50. Call usbc and make sure the youth status does not get violate.