A couple things...
1) Personally, I would insist on a deposit... with that comes a paper trail (contract) between you and the dealer. It solidifies the agreement between both sides.
2) I did it this way... about 30 days out, as that's the terms for my credit union holding the pre-approval open, I applied with my credit union with the number and terms I had in mind. Now that my truck is in transit, I asked my dealer to give me their best shot. Once I had that, and they did beat my credit union, I played that back against my credit union and gave them the opportunity to match. They couldn't. I highly recommend getting at least to quotes, and being in a position to make the best deal. This is a time of nearly free money, so work the system.