Hello,
I suppose you will have separate Payment Terms for invoices with prepayment and invoices without prepayment.
You will have to work with your APABer and try inserting a custom code in USEREXIT_FILL_VBRK_VBRP (Program - RV60AFZC) to check whether it is possible to change the system determined exchange rate.
But before trying the above, do you know whether the Pre-Payment is received before creating Delivery? If yes, then you can try changing the Exchange Rate in Sales Order itself, which will be copied in invoice.
Thanks,
Jignesh Mehta