There's not much to be lost by trying a physical gift card purchase, is there? Unless you never plan to spend cash on a Hyatt room ever again

Even if it doesn't end up qualifying for the credit, the cons I can think of are

- gift card compromise/funds drained/you lose or forget about the card forever
- if somehow it codes differently and only earns 1x instead of 4x even when purchased on site
- time spent buying the card, and/or at checkout if the front desk agent doesn't know how to apply it

The overall risk seems pretty low...

Meanwhile, as posted over in the Chase thread there's currently a Grand Hyatt 7% back offer (max $28 rebate) on my non-Hyatt Chase cards.