Term for a technique intended to draw criticism to an opposing view by overstating that view as your own (often emphatically)

Assuming he's doing it to incite blowback on the people who actually hold the views he's pretending to espouse, he could be considered an agent provocateur carrying out a false flag operation.

From the Wikipedia link:

A false flag operation is an act committed with the intent of disguising the >actual source of responsibility and pinning blame on a second party.

Trolling.

Your relative is engaging in a classic form of online trolling: posting incendiary statements in order to elaborate a response from others, taking advantage of Poe's Law to make his trolling comments believable.

I would call this a form of Fabricated Premise
He is stating a premise which he then either debunks himself or goads others into refuting.
In the example in the post, the premise is stated as an overtly derogatory comment about women serving in the armed forces.
It seems clearly intended to provoke either an impassioned rebuttal or a dismissive scoff.
This is a favorite of politicians: "They will tell you that Medicare must be reduced to stay solvent, but I promise you there will be no cuts to Medicare."

Sounds like satire and/or irony to me.

As I understand it, he is saying those things ironically.

Explicitly stating someone's position or reaction to something in a way that highlights its absurdity or hypocrisy is a type of criticism called satire. This is sometimes used to humorous effect.

While it may be framing it or phrasing it in such as way as to highlight its absurdity, it's not inherently dishonest in that it's essentially making the same argument as those whom he is criticizing were making, or not straying too far from what the original argument that is being criticized was saying. Sometimes this is done by making an analogous argument or by drawing parallels between the argument being criticized and some ad hoc fictitious argument being made in the critique.

If your friend is making a highly exaggerated form of the argument, or extending it beyond the intent of the person who originally made it, to the point of absurdity, and then your friend criticizes the result, that is argumentum ad absurdum.

If your friend is substantively altering the original argument and then is criticizing the substituted argument that is similar, perhaps superficially, but not in fact the same argument the party being criticized was making, well then that is a straw man argument/fallacy.

It is also possible your friend may have just been mocking someone or their position. Mocking, ridicule, or derision is different than irony. Irony is a statement that, when taken in context, may actually mean something different from, or the opposite of, what is written literally; the use of words expressing something other than their literal intention.

One aspect of your relative's argument technique is reductio ad absurdum. To prove a statement by reductio ad absurdum, you posit, for the sake of argument, that it is false, and then show that this would imply a consequence which is absurd. The point is that the only way this absurd consequence can be false is if the original statement is true.

There is a difference, admittedly. With reductio ad absurdum, you reveal your argument technique, and say to the effect of "suppose, just for the sake of argument, that this is false...". Your relative's twist is to pretend that he believes the statement is false. The ultimate intention --- to convince the audience that it is true --- is the same, though.

A few of the other answers have mentioned irony but I'd like to add sarcasm more specifically.

Sarcasm is the use of words usually used to either mock or annoy someone, or for humorous purposes. Sarcasm may employ ambivalence, although it is not necessarily ironic. Most noticeable in spoken word, sarcasm is mainly distinguished by the inflection with which it is spoken and is largely context-dependent.

[Wikipedia]

I think the key is that the words are used to mock or annoy someone who 'is on the side of' the point of view which is ironically (and sarcastically) stated. If the words were spoken you might hear the sarcasm (tone of mockery) in the inflection, but that tone is lost in the written communication which, in some contexts, might make it difficult to discern if the speaker holds or opposes the point of view which is represented. This could lead to annoying people who are 'against' the point of view which is stated also. Annoying all parties might be a bonus for an especially sarcastic person!

You could say he becomes a Caricature of the people he wishes to ridicule.

exaggeration by means of often ludicrous distortion of parts or characteristics

https://www.merriam-webster.com/dictionary/caricature

Perhaps you're thinking of Hyperbole

Hyperbole is the use of exaggeration as a rhetorical device or figure of speech. ... In poetry and oratory, it emphasizes, evokes strong feelings, and creates strong impressions. As a figure of speech, it is usually not meant to be taken literally.

Or from Literary Devices

Hyperbole is a figure of speech and literary device that creates heightened effect through deliberate exaggeration. Hyperbole is often a boldly overstated or exaggerated claim or statement that adds emphasis without the intention of being literally true. In rhetoric and literature, hyperbole is often used for serious, comic, or ironic effect.