How did they even get your AA number, even for the 500 miles?
As the person quoted above implies. If you look at your folio I'm going to guess your AA number is the Frequent Guest number attached to the folio; not your Hyatt number. Not sure how that happened, unless you gave it to the hotel as part of making the reservation or checking in, or, you are a repeat guest and have giving them the AA number before and that is what was on your local profile at the hotel.
What I don't know is how much influence the World of Hyatt program folks back at the mothership have over the indirect-relationship brands. (But apparently less). The pushback from the hotel may have been they reviewed the folio and had purchased the points requested. (Which is defensible if your AA number is what is on the final folio).
Trust no one. I always ask the front desk if they have my GP# on file when I check in and double check to see my GP# printed on the bill before I leave.
Also good advice. The one thing your Hyatt card is still good for is to hand to the FDC with the payment card. It is a prompt to them to check what loyalty points number they have on file. Seldom ever wrong anymore at a mainline Hyatt in the US with a system reservation made online. But I would think dropping the Hyatt card of some benefit at an affiliated brand elsewhere.
Not to mention checking for that like any other error when you review the running folio on the TV (and I know that is a presumption that the hotel in question has that feature available). It is much easier to fix the Hyatt number (and any posting errors) as you go rather than in a hurry as you are checking out.
Which brings me back to what a much older and more-experienced business traveler told me years ago when we were on a problem-solving trip I had arranged to see multiple clients. Which roughly translates I was trying to see too many people about too many things in too little time, so decide what is the one thing you are trying to accomplish above all others and don't mix apples and oranges. It sounds like you may have hit the fruit salad stage and time to call it a cheap learning experience.