Thus, if one is praying at a minyan where the chazzan's pronunciation is less exacting than the congregation's, should one answer Amen?
Why/why not? If so, is it for the same reason that he shouldn't be chazzan in the first place?
As of now, this is theoretical, but my current situation makes it very possible that I might end up with such a problem.
That's not the correct interpretation of the source you brought. There's no such abstract concept that the chazzan can't lead a congregation that distinguishes more letters than he does.
Rather, it is inappropriate to appoint in permanent fashion a chazzan who cannot properly pronounce the words. (SA,OC 53:12). See Mishna Berurah there who points out a view (Pri Chadash) that if there is no more fitting chazzan, even if he pronounces the hei and ches equivalently, he may be appointed. Note also that in the Beiur Halacha there, he says that we should also be stringent with this on a non-permanent basis.
Note also that the Taz in SA, OC 128:33 says that this is not considered a disqualifying factor for a chazzan or kohen (this rule also applies to a kohen duchaning) because רוב עמינו אין יודעים להבחין בזה
