Short of using your own cash flow to speed up the payment process, I can't see any way of doing it, short of having the customer paying directly into the dropshippers account themselves. Of course that's not a serious option lol.
EFT would be an option, funds available on deposit, but you will struggle to convince customers to use a payment method that offers them no protection whatsoever. Maybe you could try a two tier approach through your marketing. Offer two payment options, EFT for shipping within X amount of time, PayPal payment for Y amount of time, but even then you are relying on your supplier to do what you are offering ...... that's always a dangerous game offering something you have no control over.
Really, using your own cash up front to get the ball rolling is your only serious option.