It's probably due to the fact that you are booking on short notice, now the rates are indeed quite high. Even on 3rd party websites the savings are minimal.
The period I'm looking at is at the end of August. Still some time to go and in the summer holiday period.

I've never booked through 3rd party website so I'm not sure about what happens with the stay credit. I think I've read here that most bookings through other websites won't earn anything. The T&C states: In order for a Hyatt Gold Passport member to receive Hyatt Gold Passport points, the bill for charges must be in that member's name and member must personally pay for all charges. And you would normally pay the 3rd party and not Hyatt.

For my booking I haven't decided yet what to do. My reservation can be cancelled so I'm flexible. Probably one night through Hyatt with a BRG and the rest through Hoteltravel.
But I think I will first contact the hotel directly to see if they will match the Hoteltravel offer for free breakfast. I could have breakfast in the lounge but prefer a full breakfast.
If they are willing to do that I'm fine with paying €127 a night with the FreeTime offer.

Although I feel that the Hyatt website should at least be somewhat competitive with the 3rd party offers, now the differences are way too high and the BRG doesn't seem to work.