Sandpaper is ok but it's not a plane. Often you'll get uneven abrasion of the late and early growth. As long as you don't expect invisible glue joints sandpaper on flat substrate will do.

When I have to lap and dimension a thin small piece of metal, a spacer for example I'll glue a handle to it, just a stub to hold onto. Or I don't but it's easier on the fingers to do it. Starting with a file (a large one) resting on bench, propped up on my chest while sitting is usual, or whatever, drag it across- progression of finer cut files to progression of sandpaper. For scales or to avoid purchases you could use paper only.

Since the scales are not very long it is probably best to turn the plane over and carefully drag it over the blade, you can still snip the ends with wrong pressure point, using the plane in the normal fashion it is very easy to snip, and never get the thing flat.

It's also rather easy to get your finger or fingernail involved in the cut, I usually don't but it's not a bad idea to affix a small grip with white glue, rice glue, double sided tape maybe... something easily reversed cleaned off. you can push or pull it over the blade with that, while the other hand applies some directed down force. easy as pie.