I figured this out. I went back to my 9.0.4 website for this client and if the first shipping method radio button was selected, the method didn't show in the basket items on the payment page. So I compared files and AC had added a method in the Checkout Controller to assign the first shipping method.
// AS WE ARE DISPLAYING THE SHIPMETHODS AS RADIO BUTTONS SO ALWAYS ASSIGN FIRST SHIPMETHOD TO A SHIPMENT
if (rateQuotes.Count > 0)
{
shipment.ShipMethod = rateQuotes.First().ShipMethod;
IBasketService preCheckoutService = AbleContext.Resolve<IBasketService>();
AbleContext.Resolve<IBasketShipmentRepository>().Save(shipment);
preCheckoutService.Recalculate(AbleContext.Current.User.Basket);
}

This didn't work on the site because the shipping methods had been filtered out it code. So I moved the above code below where the shipping methods are filtered and the issue is solved.