Manchin refused to caucus with the Republicans because he didn’t want RINO McConnell to be Majority Leader. In other words, the RINOs in the Senate preferred to be in the minority rather than finding another leader. Now think about that!
I get the sense that McConnell and the Senate GOP do not especially like Manchin and may prefer to let the Democrats pillory him enough to drive him to the GOP or for the next election to deliver enough GOP seats to make his vote unnecessary. Like most small and select clubs, there are times when Senate internal politics makes as much sense as a pot of crabs all trying to climb and get on top of each other.