A word that brings up an idea that is the opposite of what is said

I don't have a single-word answer for you. Well, ok... I do, but it's pretty darn obscure.

So the answer that comes to my mind may not be what you're looking for.
Generally one way of pointing out the insincere nature of instances like this is the pre-pend the word backhanded.

"Backhanded compliments" is the most common usage of this. (Perhaps the initiate phrase of such usage? Comment if you have research to add.)

But, "backhanded apology" and "backhanded comment" appear in general usage as well.

Some usages are more obvious than others, those tend to be the intentional ones (note the "Especially: Sarcastic" in MW). But the subtler insincerity of subconscious "passive aggression" speaks just as backhandedly as sarcasm ever could.