Oh, I can see how that would quickly become a problem!
Can the sewing machine outer casing be removed by unscrewing it? If so, that might allow you to get at the monofilament thread and unwind it. However, if the sewing machine is computerized, you have to be extremely careful with the cables and computer boards when you take the shell off.
Just had a thought - I wonder if you could use a fine crochet hook to catch the thread and pull it out that way?
If you have another machine that uses the same size bobbins, you can wind them on that machine. That's a bit of a pain, but if nothing else is wrong with the sewing machine it's a good way to get around the problem.
Good luck!