Just be truthful, tell them you don't have a green card all you have is your work permit and social. Dealers work with many lenders, they'll make it work and get you financing with a less stricter bank.
__________________
Self filed AOS │Apps Received By USCIS - 3/18/19
│Biometrics Done - 4/11/19
Interview Scheduled - 4/24/19│Interview Date - 5/31/19│AOS Approval - 5/31/19
Permanent Resident Card Received - 6/8/19