Well, I still don't know if it works. But I did talk to a concierge at length.

She explained that Hyatt Houses technically don't have upgrades. So they should call them and let them know they are applying it and have them block the suite. If you book on points, it will not change on your side.
Whether or not the front desk sees that you are supposed to get a suite depends on which system they check. Apparently there are 2 systems, that in this case will not be the same. One is a reservation that WOH can adjust and the other is the one they use. They are supposed to look at "Reserve" instead.
There is a code WOH uses but the hotel people won't understand, because they don't use such codes.