@Ealfardan, what you are describing is Trust accounting - to give a small example.
Real Estate - the contracts are between the Seller and the Purchaser. The purchaser deposits the required funds with you (a service provider) and you make the required disbursements.
There maybe a sum left over which is your fee.
My first point is this - these deposits and disbursements should be via a separate bank (trust) account, not a bank account belonging to the business. It is unclear from your comments if you are operating with separate business / trust bank accounts.
If you are operating with separate business / trust bank accounts then my suggestion would be to operate the Trust accounting via a separate Manager Business from your main business. This way you can customise the documentation so that you aren’t being conflicted with Invoice and credit/debit note terminology.
If the Trust account holds funds which are your business fees, then your business would invoice the client but be paid from the trust bank account.
An advantage of this separation is that the client receives two statements, one showing the deposits and disbursements and one showing your fees and services and their payment. Trying to have one statement combining the two would be nightmarish as you actually have two identities - the business and the trust.
However, if you are operating with only one combined bank account (yuk) then Special Accounts should be used to separate the trust transactions, but as you have noted, you won’t get the statement look, but an export into a “master” spreadsheet could achieve something similar.