I'm sure what you are suggesting gets done by many every day of the week, and inn essence it is drop shipping. It's juts that your drop shipping without their authority.
Your success would come down to a couple of key points, being your market firstly. Of course you would need to be marketing where you were not directly competing against your supplier because you would not be able to compete on price.
Shipping time would be the other big one, you have to wait for them to process your order, then flip it to the final customer. You really are going to have to have that running smoothly or customers will burn you alive for slow shipping.
Have you considered contacting the other business and suggesting a drop shipping arrangement with them. Business is business, and just because they don't advertise themselves as a drop shipper, it doesn't mean that you can't come to some sort of an arrangement whereby you would list their products and have them blind ship direct to your customers.
Like I said, business is business and if you're selling their products, then you're putting money in their pocket.