That's not right: the prepayment requirement does not mean a guest can not change a method of payment.

I occasionally reserve rooms for the overseas guests who typically pay with cash at checkout (to avoid a royal PITA of reimbursement with currency rates additional cc fees related to foreign transactions etc etc).
I'd stop patronizing a hotel in an instant if they were to refuse to accommodate the payment switch.

Perhaps I've been lucky but so far I haven't had any problems with switching - this year it worked at BOS Harborside, Tahoe, OC and probably a couple more.

I'd make a list of the rogue hotels that refuse to switch and bring this to Hyatt's attention. The lack of cooperation from those properties undermines the idea of gift cards and is likely to alienate people who expect to pay with GC at Hyatt. I don't think Hyatt wants to deal with guests claiming that their GCs were refused - that can only lead to negative publicity.