Back in the olden days when we wuz young we used to wind coils with litz wire
- which resembles the wire you describe - when soldering we didn't take the
trouble to bare each individual strand, but rather used very fine wet and dry
to gently clean the wire treating it as if it was one conductor. Worked
pretty well.

So, you could try this easy/lazy/can't be bothered way and see how it goes

David