640 or higher you should be fine. the higher the score the lower the rate. another important factor will be debt to income ratio. some lenders will only lend up to 17% of gross income or about 40% total diti.
Your DTI will be a huge factor as well as any bad marks on credit reports. I personally wouldn't assume a car loan of that amount without a score of 740. Hope you do your research on this car before you commit and best of luck to you!
You SHOULD be fine if you fall into the DTI ratio as listed. Also, a $7k down payment will be hard for many dealers to turn down, especially considering your 700+ Ficos.
Maybe what you can do is print out your reports/ficos and go to a dealer. Have them look at your income and reports and mention you will gladly allow an inquiry if they realistically feel you can qualify. Mention what you will offer as a down payment (heck, start with $5k and see if you can save) and see where it goes. You are the one in control, don't forget this.
Considering how many dealers NEED to sell product, it's hard to turn down $5k + 700 ficos for the sake of it. Let alone the potential $7k you would put down. Worst case, get a certified used BMW and eventually trade it in. I hear their financing is easy to work with for somebody in your position. Do a search in this forum to learn more.