No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
Capital One's Navigator will give you an amount that they will finance for. Putting in a lower amount may change your odds of if they will approve you, but I would put in for the larger amount that you would be approved for unless you find a specific car you like through their tool. Having a larger amount will give you more options.
Taking your own financing will help a ton, just make sure you let the dealer know that you have your own approval already and that you DO NOT want them to pull your credit. Having an approval from Capital One is not a guarantee, but from what I've seen on here and through personal experience, there normally is not a problem with them not actually approving the terms from a pre-approval