This is how I would do it, if there are still a lot of empty seats when you get to the airport. Also, check the app or website for the standby list, as you can assume all of those will take up an empty seat.

Smile and say hello to the gate agent. Say that while you were expecting to check the car seat, you noticed there were a lot of empty seats on the plane and wondering if it were possible to take the car seat with you.

Be prepared for the answer to be "No", and say thank you...sometimes a "No" for operational reasons turns into a "Yes" later if you're polite to the agent. But if you get the right agent, in a good mood, and he feels he can accommodate you, be prepared that your family may be reseated as well, and they may ask you to wait and board last.

Whatever happens, remember you're asking UA for a big favor, as the rule is that you have to pay for that seat. If you act accordingly, you might get what you ask for.

Good luck!