I think the mutation that grants something like immunity to (more) mutation is Homogeneity, though how that plays with what is already a three eyed mutant is anyone's guess.
Given that in 40K the Webway is explicitly falling apart I do not think it is reasonable to count on it to just resist the literal tide of entropy that is the warp forever. Put it this way the Webway in canon has existed in something like it's present form since just after the War in Heaven sixty million years ago, it has been without maintenance for ten thousand years, that is 0.016% . It ain't doing so well in the present.
Homogeneity suppresses magic powers in general.
Immunity to warp mutation can be gained with the power Immunity, which means you aren't affected by the thing you're immune to. We just pick warp energy as that thing.
Balefire is an explicit example of something you can be immune to (basically evil green magic comic book radiation in Werewolf), which is a pretty good analogy for raw warp energy, as it's a similarly damaging, mutagenic energy.
On the Webway, I think most of the damage was done by the Fall or by subsequent daemonic intrusion, and the Emperor has non-psyker soldiers who can kill daemons (blanks).
And as I mention, the Dark Eldar can maintain the parts of the Webway they control with technology rather than psyker powers. The Emperor could easily be planning to steal that from them and give copies to his own people.
The Dark Eldar don't need psykers to maintain the Webway, so it's not unreasonable for the Emperor to assume that he doesn't either.