What kind of motor would I use to drive a mass back and forth in a linear direction?

A linear stepper motor would fill the bill. Not cheap and compressed air is needed for the shuttle clearance.

The heart of such setup could be a micro controller, programmed to make the servo respond as desired.

Precision locating and acceleration/deceleration can also be programmed.

2 Oz. is not much mass, a 1/4 inch is quite short, so even a smaller servo could be employed.

I designed a conveyor system that had to move nearly 3 Lbs. at a controlled velocity over 10 inches, rapidly returning and repeating the process on cue from sensors, repeated up to 4 times per second.

There are multitudes of servo configurations and sizes, just locate one with the capacity and power to do the job.

So, it can be done, even mechanically, but less control.

How about an electric model train , The old steam engine kind. They have a rod to connect the engine wheels to the simulated steam engine. It would go back and forth with ample power and you can vary the speed by adjusting the electricity. (?)