I really doubt that the fix is killing your prints, that is unless you have some contamination that's turning it into farmer's reducer. I use rapid fix at film strength and have no problem with bleaching. I'm going to venture a guess that what you are seeing is what Tim Rudman refers to in his book as "fix up".
Lith printing is very paper specific and some of the papers listed by Mr Rudman no longer work the way they did when that book was published. The Kentmere/Luminos papers were amazing before they removed the cadmium.
Another thing to consider is the contrast and density of your negative. I like contrasty negs for lith printing but if your negative is too dense you will get into reciprocity failure with the paper. Try a fast paper like Oriental VC.