Nope, you don't get anything when booked through an Online Travel Agency (including Chase points). Doesn't matter whether you paid cash, points, or a mix.

https://help.hyatt.com/en/hyatt-term...att-terms.html
Upon check-in at a participating Hyatt hotel or resort for a stay booked at an Eligible Rate ... a Member may choose to earn one of the following: (a) points in the Program ...

An “Eligible Rate” is defined as any room rate for a participating hotel, resort, or other property that is published by Hyatt (including rates found on hyatt.com without the application of a discount code, and Hyatt’s “Standard Rate,” “Volume Account Rates,” “Senior Citizen Rate,” and the monetary portion of any Points + Cash Award (as defined in Appendix B) or, for Miraval properties or Oasis Homes, published by Miraval or Oasis, respectively.

“Ineligible Rates” are all rates other than Eligible Rates ... rooms booked on third-party online retailers (such as Expedia.com, Travelocity.com, Hotels.com, Booking.com, Priceline.com, Tripadvisor.com, Inspirato.com, home-sharing sites, or any other third-party online/mobile travel portal or bookings made directly on Google.com and TripAdvisor Instant Book), traditional wholesale rates (examples include, without limitation, GOGO Worldwide Vacations, Pleasant Holidays, Tourico, and Hotelbeds), airline crew rates, airline employee rates, travel agency employee rates, media rates, Club at the Hyatt discount stays, Hyatt employee or employee family or friends-and-family discount rates, airline interrupted-trip vouchers, or contracted rooms.


So basically, you get no status benefits, night/stay credits, points or anything if you book your hotel online through anything other than a hyatt website.
(Note: There is anecdotal evidence that AmEx FHR and other more "full service" travel agencies do count as eligible nights, but I don't think that's ever been confirmed with Hyatt that it always will count)