No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
Have you tried talking to another dealership? If you have never been late and your income can support the car note what the dealer said makes no sense.
Edit to add: I believe from reading your post again that you are still paying on the first car. Is it possible that you have too much negative equity in the loan that they can not secure the financing?
I think he is actually trting to purchase a second vehicle, in addition to the first. If income will supoort it I don't see why it should be an issue. Maybe look at another dealer or brand? Local CU?
As long as your income can support both cars it won't be a problem.