Ok, so this is based on my experience as a hobbyist woodworker and from fixing up scratches and orange peels on my car. You will need:

Fine sand paper with grits 600, 800 and a 1000. If your scratches are deeper, you might have to go down to 400, but it would be better if you didn't (don't go below that — will be too hard to recover on plastic). If you'd like a much smoother finish, go up to 2000, but it isn't necessary.

Buffing compound.

Buffing cloth with microfibers.

A can of clear coat spray.

A steady hand.

You can easily get the first 4 from an auto parts store (e.g., AutoZone in the US).

Caution: This will generate fine dust and the fumes from the spray can cause irritation, so use a dust/vapour mask.

Start with the lowest grit sandpaper and gently sand the roughed up area. Be steady in your strokes and use long strokes instead of short, rapid and jerky motions. Of course, "long strokes" is kind of ambiguous in a mouse that's only 4" long and with a scratch length much smaller than that, but the important thing is to be steady.

Don't take too much material and don't apply pressure. Let the grit work on the scratch; not your strength. Once it starts getting smoother and you don't see any visible improvement, move higher up in grit. Repeat the same till you finish the highest grit. Remember that the plastic layer isn't very thick and possibly contains sensitive piezos underneath, so do not dwell on each grit for too long.

When you're done, you should see a smooth, matty finish on the surface. If all you want is smoothness and don't care about gloss, I'd say stop here and move to the next paragraph. Else, spray about 2 light coats of clear spray. Sand gently with the highest grit in between coats (allow each coat to dry completely) and wipe with a microfiber cloth before the next coat. If you're worried that an additional film layer might affect the touch sensitiveness, by all means, skip this step.

Finally, apply the buffing compound and work into the area with a buffing cloth. The compound serves to fill in the ultra fine scratches that the grits won't get and brings out a nice finish to the piece. Use an orbital buffer if you have one, but hand should be sufficient.

Good luck, and hope that works for you.