score:3
The issue here is that the insurance company is going to make sure they don't get over charged for repair work, or charged for work that never happened. That sort of thing unfortunately happens when people hear insurance companies are going to pay for something. This means that the insurance company is going to want to see several estimates for the damage repair (and compare them with their own figures), and receipts that show that the work was in fact done.
However if you pay the hotel up front, they have no way to make the hotel give them the documentation they require, and thus ensure they are not overcharged. If you just pay up front, they may decline o repay you, or they may pay you the amount they would estimate to have the damage fixed, which may be less than you paid the hotel.
Unless the amount the hotel wants you to pay is clearly small enough that you would pay it just to make the problem go away, you should also be asking for repair estimates and receipts before you pay anything. That means not paying before you leave.
I would hold the line on this. Give the hotel your contact info, and assure them that you will pay for the repairs. Tell them you will need repair estimates and receipts before you pay, and that you want to do this through your insurance. The hotel has no right to demand that you settle for damage immediately, and no way to force you to do so, unless they can produce the documentation immediately (which would be very suspicious).