Suggestion for controlling a stepper motor on batteries

I am working on a project to add automation to a classic holiday decoration. I have an Ernest the Balancing Bear. It is a toy/decoration from back in the day. Essentially it's a self-balancing bear on a wheel that rolls on a tight wire. You affix the wire to a wall and pull down on the wire and the bear rolls towards you. What I would like to do is fix one side of the wire to the wall and then move the other side up and down with a motor and Arduino. I would prefer this be powered by batteries.

My thought was to use an Arduino Nano with a Stepper motor and driver. I could use the standard stepper library. I would attach a small arm to the stepper and attach the wire to the arm. I could spin the motor back and forth to half and negative half, forcing the wire up and down. I have attached a sketch of the concept for reference.

Is using a stepper with an arm a good approach, or can you think of something different? If a stepper is the best route, how do I go about figuring out the size needed and then matching a power source? Is using battery for this realistic?

I have attached a motor spec of one I have on hand if it could work.

Stepper motors are not good friends with batteries due to the constant high current draw (they're very inefficient).

Unless you have a case for wanting exact position and speed control, you could probably achieve the desired effect with a geared motor driving a screw, the screw moving the left-hand side up and down. With the help of a motor driver (e.g. the TB6612FNG) you have very easy control over speed and direction.

Thank you for the thoughtful suggestion. I happen to have a motor driver so that’s a good start. What kind of component would I use for the screw and the part that goes up and down which connects to the bear wire? Is there a name for that kind of assembly?

This sort of thing may be of interest:

You could probably build something pretty simple with a length of threaded bar attached to the wall either end via a bearing and the decoration attached to a nut on the threaded bar which as the threaded bar is turned moves up and down.
A simple DC motor controlled via a H Bridge could run from a battery (or at least low voltage to avoid any safety issues).
Depending on the accuracy of positioning you require a simple switch at each required position along the threaded bar may be enough or a sensor which detects each full revolution of the threaded bar to count the amount of travel?
For inspiration try searching eBay for "pillow block bearing", "leadscrew", "linear rail", "h bridge", "geared motor", "shaft coupler", "limit switch"
BTW - Don't forget to include a fuse of some kind in case the motor gets stuck and starts to overheat etc.

Thanks all. I’ll do some research and see what I can do.