How about melting the thread? You could touch it with a hot iron. Hopefully, that will melt it. Practice first to see just how hot it has to be to melt without burning the fabric, and/or melting into the other stitches. I would back stitch a time or two THEN hit with hot iron. When using fishing line, a lighter flame on the end keeps the knot from untying.
Know what you mean about Sulky invisible thread. I had problems with it too. It kept breaking, even after I changed needles, adjusted tension, etc. I took it off, took hold of the end and pulled it off the spool. I kept getting pieces of thread. I looked on the thread on the spool and could not find a "slash". The only thing I could figure was that they put end pieces on the spool as it was not continual. Never bought Sulky again! I love the Mettler invisible. It is very thin and does not leave pokies as it is soft. Not sure how it would work for the strength that you need though.