I love your idea. Make it $1000 to enter the lottery:Non refundable fee of $500 or more. Even if that's not idiot proof, atleast legitimate candidates will have the satisfaction that idiots were hurt.
That fee gets applied towards your application, if you're eligible.
1. if you're selected and qualified, $1000 counted toward your app fees.
2. If you're not selected, you get $500 back
3. If you're selected and not qualified, too bad, you lost your lottery ticket.