OK, this is an odd coincidence. I thought Hyatt didn't do a discount as part of their BRG, but I hadn't looked into it in ages, and found the page tonight. Lo and behold, they offer 20% off! And now I find this only changed in the past couple of days.

The inclusion of free membership sites is pretty extraordinary. Priceline offers me some insane discounts from time to time if I'm signed in to my profile. I can't use those with Hilton, but evidently I now can with Hyatt.