Is a bank allowed to finance late fees due on a prior loan?

Check your state law. But IMO, uncollected late fees are part of the payoff balance to be paid, or refinanced.

I'm looking at OCC Bulletin 2000-20 and it says:

"Institutions should adopt and adhere to explicit standards that control the use of extensions, deferrals, renewals and rewrites of closed-end loans. The standards should exhibit the following:
-The borrower should show a renewed willingness and ability to repay the loan.
-The standards should limit the number and frequency of extensions, deferrals, renewals, and rewrites.
-additional advances to finance unpaid interest and fees should be prohibited.

Late fees should not be financed. If the borrower cannot pay accumulayed late fees they probably cannot repay the loan either.

Second the bank shows income (proceeds applied to late fees)that it has not actually collected and bank income is distorted. Thats what examiners will ding you for.