I don't think any of the Japanned sewing machines actually are French Polished. I was tought it years a go by a furniture restorer (carpenter), he did a very thorough job, sometimes retouching, sometimes full refinishing of veneers.
To get the exceptional sheen on wood they use an especially prepared cloth (easily put together) various staining and "wetting" of the wood before the lacquer is applied, and then sprinkle a fine dust of chalk dust on the first couple of shellac applications (to fill out and polish out any pores in the wood). The remaining layers are done with out the chalk dust. The steps sounds like it's much more fuzz than it really is.
As mentioned, I don't think any Singer machines or other brands actually are french polished, but done in a more simpler application method suitable for large scale production. Singer sprayed the shellac on, but I have never found any info on the details. I have seen aerosol spray cans with shellac in stores, rather expensive compared to the flakes you dissolve in a jar of alcohol. If you have spray equipment it can be used. There are ready made tins of shellac available too, but around here the flakes are most economical and exactly the same thing. (The lowest grades and prices can be less puried, and you might noticed traces of something in the bottom of the jar, some filter it out through a nylon stocking). You don't need the blondest most purified types, the medium to darker hues turn out transparent and are said to be more durable. With a few odd exception matching it, I have never seen a finer or deeper gloss than with the french polish technique.
For the spray method, it should be doable. The cloth application is easy tough, the making of the bundle is easy too; soft cotton wrapped around a bit of the same type of cotton. Make sure the surface is even; to prepare it; soak in shellac, then add one or three drops of clear sewing machine oil to make in run smoothly over the surface. Either method should give a flawless result and it's not a lot of work compared to other lacquers.