Have you looked at transfer wise. It looks interesting and may be an option. They say their fees are lowest.
Related: Paying a Poland-related bill from the United States
I have the account number for the bill I need to pay - however, I live in the US. Is there a way to pay by credit card or paypal and then the funds would get transferred to the bill account number?
If it's a bill from a shop or hotel, they should be able to take it off your credit card, you just need to authorise them. If it's fine or utility bill, then it might be difficult.
In any case you should speak to their accounts, if they want their money, they should make it possible.