If you strap the sled to a bench or use excessive counter weight to eliminate the recoil (as supposed to reduced), something has to give ... the weakest link, i.e., rifle stock, etc ...
Have an effective muzzle brake installed instead.
+1 with what FEENIX said. Do NOT strap it down.
I've never had trouble with it sliding too far on plywood, concrete, or planks.
Maybe you could try some new flexible rubber on the bottom?
I shoot all kinds of rifles off my sled, from 340 Wby to 22lr & I've never used any weight. Sled slides about 1-1/2" back with the bigger boomers, & not at all with the lil ones. I use mine mostly just for a rest, but the added bonus of recoil reduction is great with the big cartriges.
Maybe you have too much weight, so the muzzle jumps? I dunno. I've never had that problem, but again, I've never used any weight at all.