It's "Congolese", and I think if adjectives were required to distinguish nationality, one might use "Democratic, Republican Congolese" or perhaps "the Congolese of the Democratic Republic."
I'm not sure if just "Democratic Congolese" works. And "Republican Congolese" or simply "Congolese" is probably best reserved for the smaller country bordering on the west, since it's named Republic of Congo. But really, I think citizens of both countries can pretty much share the title of "Congolese" in terms of ethnicity, wherever applicable. Their current borders and formalized names have only persisted since the late 90's.
