The trouble I have with this advice is that every specific example or suggestion on how to change my message has ultimately boiled down to "soften the message enough that the recipient can feel okay with ignoring it."
But that defeats the whole purpose! If someone is actively entertaining a cognitive dissonance of some kind, "softening" the message merely gives them the "out" they need to continue to hold the dissonance!
Unfortunately, because I've been given this advice so frequently, and because it always ends up being "just don't challenge people", it comes across as rather condescending. I'm sure that's not your intent, but next time, please consider that maybe I've already done that analysis a hundred times over, and still reach the same conclusion.
Perhaps I'm just stubbornly wrong! But I really don't think the issue is that; I think the issue is that we've made it socially unacceptable to call people out for being inconsistent. Just look at what the typical responses to such a thing are: whataboutism, radical generalization, ad hominems, retreat to an echo chamber, the classic gish gallop of tangentially-related things, etc.
Perhaps what we've really made socially unacceptable is the admission of fault?