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I'm pretty sure you can have your dad take you off the card easily for a short time since you are just an AU, do you have any other cards with a long history?
If your FICO is good and you are not on the cusp of a critical number, have him remove you ASAP and get with the CRAs to get it removed. As I recently mentioned in my post...Underwriters are idiots, plain and simple. There is no need for you to have to pay it all off to close. Your Dad's recent statements should suffice to show what is the minimum due and that it's being paid timely. Best of luck! And to you underwriters..."Bleep Off!"
It is senseless they make you pay off, because you are not responsible for the payments in the first place. Also, they should not ask you to pay it off anyway, unless the minimum payments are cutting in to your monthly mortgage payment, such as if you are borderline with the DTI including that payment. But again, you are AU and they should not ask you to pay it off to begin with. I guess it depends on the lender.
Is it possible that the AU agreement makes the AU jointly responsible for the full balance due? If not, then get yourself off the account abd build your credit profile on your own tradelines. Generally, an AU is not responsible for payment.
your UW is an idiot that needs to find a new career. Tell your LO your not interested in doing business with them. It's not your debt and you aren't responsible for it. Also I'm sure the $25 payment doesn't mess up your DTI. Fire them, find a good mortgage company.