Their position seems to be... it depends, lol. But they prefer CPC.
Personally, I prefer to use CPC because the implied accusation of ethno-supremacism is front and center in much of western news media. It's the Chinese Communist Party -- the emphasis is not so much on "Party" as the CPC has stated, in my experience, but rather on "Chinese".
In fact, I'd wager the Transatlantic Empire would rather the CPC drop "party" from their name altogether, as it indicates a participatory role in democratic processes. "Chinese communists" would probably be far easier to effectively scapegoat.
CPC also rolls off the tongue easier. Also serves as sort of dogwhistle for people who are aware of all of this. I see no advantage in using see-see-pee.