All major DM candidates also have multiple interactions: that's the WI in WIMP, for instance. In fact I don't know that anyone is seriously proposing that dark matter is just bare mass with no other properties - aside from the practical problems, that would be a pretty radical departure from the last century of particle physics.
No interactions have been found, despite a lot of resources put into the search. So currently all dark matter particle theories apart from "non-interacting" have been falsified. And non-interacting theories are probably unfalsifiable.
Radical departure may well be needed, for other reasons too.
