Bringing an Extra Kid to a Hotel Room Beyond Capacity—Anyone Have Experience?

I wanted to ask if anyone has experience discreetly bringing extra people into a hotel room that technically exceeds the listed max capacity. We're looking at a Hyatt hotel, and the room we're considering is just a standard hotel room—about 300 square feet with two twin beds. Not a resort or an all inclusive.

The max capacity listed is 2 adults and 1 kid, but we are a family with 2 adults and 2 kids, and the kids are 10 and 7.

I already called the hotel, and they said they can’t allow the second kid, so I’m trying to figure out if this is something people have gotten away with before. I’m also using Chase Points for this stay, so I really have to keep it within the Hyatt hotel chain.

Has anyone done something similar, and did you run into any trouble with the hotel staff? I am hoping when I check in and the receptionist asks for our passports, I will just give 3 of them, while they are outside the hotel. Were they strict about enforcing capacity limits, like if they spot us later on in a family of 4.