Can’t you just accept the immediate cash payment, but make sure that it’s recorded correctly in your accounts.
I don’t know where you live, but the tax law I’m familiar with doesn’t care if the customer received a bill in the post or even if they recieved a bill at all. The critical thing is that all payments received within the tax period are declared.
You could still produce a copy of the bill for your records and even a receipt for the payment if it makes you feel better.