However, imagine that there's a really strong social stigma against asserting that murder might not be bad, to the point of permanently damaging such a person's reputation, even though there's no consequence for making the actually stronger claim that all morality is relative. The relativist might therefore see the critic as the one who is disingenuous; trying to leverage social pressure against them instead of arguing on the basis of reason.

But the reason people have stigma against asserting that murder isn't bad is because they (presumably correctly) think that moral opposition to murder prevents a lot of murder, and so people who don't think murder is bad could potentially end up murdering others. Insofar as they make an exception for relativists, it's presumably because they think the relativists either haven't realized that murder disproves their position, or they think the relativists know of something that makes murder an exception to the general moral relativism.

If either of these conditions apply to the moral relativist, then bringing up murder is helpful because it helps highlight that the conditions apply. If neither condition applies and the moral relativist doesn't believe that murder is bad, then bringing up murder is also helpful because it helps discover that the moral relativist is a potential murderer who must be removed. Thus bringing up murder is helpful regardless of what case we're actually considering.

More abstractly, if we model this notion of moral relativism as "all moral claims are meaningless", then it is a statement of the form "all X are Y". Such statements ground out to the conjunction of "x is Y" over all X's, so it is always earnest to replace "all X" with a specific x. That said, sometimes it may be counterproductive to replace with a specific x, if it is complicated to evaluate whether x is Y or if x technically isn't Y but it's a weird unusual corner-case X that could plausibly be excluded in a fixed category X'. So a productive mode of engagement is to pick an x where "x is not Y" is an especially relevant counterexample of the generalization. This sure seems to be the case for x="murder is bad", Y=meaningless.

Like basically, lowering the relativist's social status isn't an attempt to use social pressure to get them to change their mind. It's just making sure that their status accurately tracks their vices (which, heck, in a sense, surely this is something the relativist should accept, since presumably the reason they want critics to be reasonable is because they believe the map should track the territory and reason is a good tool for making accurate maps). It may be that it also functions as an incentive for the critic to lie about their views, but really that's a bug (you'd rather have potential murderers say so publicly so you know who to be careful about), and if this is the function in this situation, it's reasonable for people to decide that the critic is disingenuous (as that is literally what they are).

The critic could respond that our only measure of a moral system is whether its conclusions agree with our intuitions, so we should find conclusions about which we have strong intuitions.

The relativist could respond that our intuitions are noisy and so we should use error-correcting heuristics like Occam's Razor and bounding the score impact of each intuition.

This seems trivially true. If there is social pressure that applies (especially if the conversation is public), then any utterance will be somewhat altered from the underlying true beliefs of the speaker.

It's worth reading some Leo Strauss, regarding esoteric writing which can be interpreted correctly by your intended audience but which will pass inspection by those who would punish the expression of such ideas. This is probably the standard starting point.