See if you can beg borrow or steal a Pioneer HLD-X9 or its cheaper cousin the Pioneer LD-S9. The pickup on these is far superior, they will play discs even with advanced rot with no sparklies and have by far the lowest noise floor of any LD player out there. The American CLD-99 is another good player, same pickup but not quite as good a comb filter.
But really any of these players will give an image that is so much better than the player you are using that there is just no comparison. Seriously.
If you sniff around a Laserdisc forums (Try AVS) you could probably convince an owner to lend it to you for the few days to get a super clean capture. You will be truly amazed at the difference in quality.
It would be a shame for you to expend so much effort on this only to have your source footage be a long way below par and make the final result that much less than it could be.
As my Ma used ta say - Ya can't polish a turd.