It's all smoke-'n-mirrors. Negative equity is never "hidden" and never disappears. It's either added back into the loan or paid up front in cash.
You're going to have a hard time getting to that $300 a month payment with that much negative equity. And, it's going to be very difficult to find a lender willing to add that much negative equity on to a loan and approve you. Lenders will only approve maybe 105-110% of book value on a vehicle, and that's with good credit. You're not in that situation. With poor/marginal credit, you're going to be lucky if they'd approve you with thousands down and an instant equity situation. If you're wanting to add that $4000 into the new loan, you're now asking the lender to overallow by as much as 140%. Not gonna happen. Plus, even if they did approve it, you'd be stuck in that car for the life of the loan.
Keep this in mind as well... if you take on a car loan, you must carry full coverage insurance. So, that magical $300 payment is just the beginning.
Trying to swallow that much negative equity in a $15K car isn't going to happen... not sure who told you that. If you were talking an $80K car, I'd say the chances were WAY more likely.