Completely valid concerns! Right now the payment registration window doesn't seem to have the additional fields to add an external document number and it will always show all of the open AR transactions.
It sounds like the straight "deposits" window would work better for you guys. Search for "Deposit" > Deposit.
If you haven't used this before, you may need to navigate to "General Ledger Setup" and assign a number series to the "Deposit No." field.. and you may even need to add a Deposit number series. You can do this by searching No. Series and adding a new row for Deposit.
Once this is ready to go though, in the deposits window just enter the header info, bank account, total amount of the grouped deposit, and then enter your cash receipts in the line items section. Remember to also Apply each cash receipt to the invoice by going to the ribbon under "Functions"> Apply entries.