As you can probably guess from the quotation marks, the hang up here is the word provable. The thing is, you can't just prove your financial need in any old way. You have to prove it their way, with FAFSA. The trick with FAFSA, though, is that all it takes into account is how much money your parents (and yourself) make in a given year. The amount of proven financial need you have, then, is based not off of how much money is being given to you for your college education, but how much money your parents could be giving you. And this is where the rub lies. Because it doesn't matter if your parents are millionaires or dirt poor - if neither of them are giving you money, you're still equally broke.
So, how then can you get scholarships that require financial need? By being upfront about your problem. Apply to a scholarship that says you must have financial need, but does not give specific requirements for how much need you must have. That means looking for a scholarship that does not require that you have a Pell Grant or a specific FAFSA number. In your essay, especially if room is given to explain your financial status, tell the reader that while your FAFSA number says you have no need of aid, your actual status is different. Then explain why. Tell the reader that it is even more important that you receive their scholarship because it is one of few that does not have specific qualifications for financial need. And then move on to something else. Do not sound bitter or self pitying when you mention this. To make sure it comes off right, try having someone else read through the essay before you submit it. If all goes well, you'll have a higher chance of getting the scholarship, or at least be considered for it. I've tested this myself with success and you can do it, too.