Unfortunately your client is correct, the time of supply is the earliest of invoice issued or payment received.

9. Time of supply

(1) For the purposes of this Act a supply of goods or services shall, except as otherwise provided in this Act, be deemed to take place at the time an invoice is issued by the supplier or the recipient in respect of that supply or the time any payment of consideration is received by the supplier in respect of that supply, whichever time is earlier.

There is a list of exceptions and I don't see where the service you offer will form part of any of those exception