Is ‘suit-wearing’ an adjective sui generis?

It's simply using two words together.

Dennis Kelleher runs Better Markets, a nonprofit that pushes for tighter regulation of American banks, something like Occupy Wall Street’s suit-wearing cousin.”

Occupy Wall Street is the name of a protest group, and the protesters generally don't wear suits.

Better Markets, on the other hand, is a group with similar aims, but they wear suits.

The phrase "suit-wearing cousin" is meant to contrast the Occupy movement's casual attire, and the unprofessional attitude that goes with it, against the formal attire, and the attitude and actions that go with it, of the Better Markets organization. The word "cousin" literally means the offspring of one's aunt or uncle, but the proverbial meaning of something closely related.

(I cannot attest to the attitude or behavior of either group.)

Your question assumes a false dichotomy between 'a full-fledged adjective' and ' a combination of two words'. The combination of a noun and a participle to form an adjective is perfectly normal; though any individual result may not be common enough to appear in dictionaries, it is neither irregular nor colloquial. Man is a tool-using animal would be the normal phrasing of an academic paper, and You mouth-breathing idiot! may be uncomplimentary but is not bad English. In your example, suited or besuited would also be grammatical, but would hardly fit in a newspaper article.

It's not a special word unto itself, but it's a very common pattern, an informal way of converting a V+N sentence into an adjective.

He is wearing a suit.

converts to

He's a suit-wearing person.

Nominally it just means that the person is wearing a suit. Pragmatically, the pattern is used to point out a feature of someone, something that the person does, that is not particularly favorable, that is usually when attempting to give a long insulting description of someone:

That cotton-picking, carrot-eating, cow-tipping, varmint

Note that these descriptors may not be literally disparaging (eating carrots is pretty innocuous), but the cultural pattern is that it is intended to be an insult.

The particular phrase

Wait a cotton-pickin' minute

was a catch phrase of Bugs Bunny in the 1950's. Here it is only used as an intensifer.