If your credit score is favorable, the outstanding card balance will likely have little or no impact on your interest rate. Student loan debt is almost always classified as non-consumer debt, it's difficult to get student loan debt discharged via bankruptcy, and therefore it's a bit less risky for the lenders. Private lending interest rates will still vary by score, but there's always an upper limit on the benefits of a marginally better score, ie my 805 score doesn't really do anything for me that a 740 wouldn't. Furthermore, paying off early would likely have little impact on your score, according to this equifax article:
Does Paying Off an Installment Loan Early Affect My Credit Score?
Edit: The article references installment loans which are not the same as credit cards, but the impact on credit is similar, except that a paid off credit card is not a closed account.
Federal Direct loans are need-based, so it would make no difference there.