You probably won't be able to hold onto the child/seat in a bad accident.
https://www.quora.com/Can-I-install-a-child’s-car-seat-in-an-Uber-vehicle-for-my-son-Will-the-driver-mind
"Uber has a choice of requesting a vehicle with a car seat. For example, after selecting your pickup location, click on UberX and you should see the option for "car seat" Keep in mind that a $10 surcharge is added to UberX pricing for this type of Uberfamily trip.
uberFAMILY provides one forward-facing car seat for a child who is at least:
(a) 12 months old AND
(b) 22 lbs. AND
(c) 31 inches.
A child is too big at 48 lbs. or 52 inches."
So not for you
BUT
"If you choose not to select this car seat option or if there does not happen to be any drivers with a car seat around you, you might want to request a regular UberX and ask if the driver is ok with you installing a car seat. If it is not ok with that driver, you can cancel the ride within 5 minutes of requesting and you will not be charged any penalty."
If it's easier for you, check to see if your car seat can be installed without the base. There should be instructions in your manual on how to install without the base.
For more information about car seats, visit
www.bestcarseats.org