If you convert this to a 120V panel, then adding a neutral is exactly what you are doing. It's the only possible way to make it a 120V panel, since 120V requires neutral to exist.
Now, your panel currently has a neutral bar, which is being misused as a ground bar because it was originally installed as hot-hot-ground and it wasn't being used as a neutral bar. Now you need it for that.
So install an accessory ground bar - the panel labeling will list which models of bar are made to fit pre-drilled sites around the panel. Move the grounds to it, emptying the neutral bar.
If the bare wire can't reach the ground bar, then obtain a suitably sized bare lug, drill a hole with a #21 drill bit NOT on a knockout, and use a self-tapping 10-32 NF screw to mount the lug to the panel.
Remove any neutral-ground bond and pick a hot wire to become neutral, and move that wire to the neutral bar. Mark it with white tape (legal since the wire is #4 or larger).
This panel actually has 4 spaces, but because 2 of the breakers would be upside down, the upper spaces cannot be used except in Canada. Also note that this panel has a Square D HOM and Eaton BR breaker. Those are not allowed in the same panel. Figure out if this is a Square D or Eaton panel, and get or keep the correct breaker.
If you need to power both remaining breaker spaces, you will need to pigtail the remaining hot wire to both lugs. The least bad way to splice is a 3-port Polaris connector of the smallest size allowed for the wire size. This will NOT be cheap and most likely your better option is to remove this panel from the wall and install a 6-space or larger panel. Spaces are cheap, so I would go with a 12-space unless you are constrained for space, then consider a CH or QO 8-space panel.