A DC gearmotor with limit switches will probably work. A servomotor has additional sensors that let you emulate the limit switches, but also do more, so if you use a suitably strong servo, that would work. It probably doesn't need to be fast at all, so a very high gear ratio would be preferable.
There's also a safety issue -- you don't want the motor to be so strong that it will choke a baby who happens to be laying on the back seat when the button is pushed...
An alternative is a linear actuator with an arm that turns the armrest, but you may not have space for that within the constraints.
There are small, high gear ratio, servo motors that can probably do what you need, such as the Robotis Dynamixel MX-106. That servo also runs on 12V, which is good for cars, although you probably also want a TVS or other kind of "load dump voltage" protection.
The benefit of a motor with gearing is that you can mount it straight on the axis of rotation, which makes it less messy to install.