No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
Are you certain which FICO models the creditors you're applying with use for decisions on auto loans? The extra scores come with the purchase. See also -- it's the very first bullet point for each product:
http://www.myfico.com/Products/Products.aspx
I don't recall how often they're updated but it's not as often as the FICO 8's.
Here's my 2 cents. Before you buy all your scores (which can be pricey for just one pull) you might want to huddle with some people here and get some general advice on how to improve your scores in a very short time. (Basically, make sure that your credit card balances are in the best possible shape before you lay out the money for that score pull.)
There's also a chance that your standard FICO 8 scores are so high now that pulling your Auto Scores would be just a waste of money.
So my advice personally is to pull your TU and EQ reports through Credit Karma (though the scores will likely not be used be the lender). That will cost you nothing. Then work with people on how to improve the reports as fast as you can. You'll be able to get free updates to your reports once a week.
When do you think the student loan will appear on your reports? Three weeks? Three months?
So the SLs when they appear will increase your profile's age a lot? Sounds like a good choice then to wait.