[David G4EBT]

As your sewing machine is a manual one, using the bobbin winder with a 4mm shaft might be a bit more 'tactile' than other methods. Turning the bobbin with one hand, and feeding and guiding the wire onto the former between the fingers of the other hand, lightly tensioning the wire as you go. 337 turns isn't very many and doing it slowly by hand overcomes the problem of jerkiness which will arise if winding at a faster speed. 0.0.4mm is 48 SWG so quite fine.

As you have a lathe, alternatively, you could mount the former on a 4mm shaft held in the headstock and rotate the headstock by hand using your left hand, while guiding the wire with your right hand. If your lathe can be run at a slow enough speed, (50 RPM for example), you could of course do that rather than turning the headstock by hand, but it's so few turns that in my view, hand winding gives a better chance of success.

If it's modern wire that you're using, it will most likely be self-cleaning/fluxing, which makes soldering much simpler.

Good luck with it, whatever you do.