I think Bloomberg has his own view of how things should be (soda, guns, diet, general nanny state). Party isn’t a religion for him, it’s a vehicle. I don’t think he cares about either Party, except as a tool for his ambition to affect and control society.
Of course you are right but this sort of belonging to a Party of immediate convenience will not play well with thinking voters (especially the real Independents) who like to at least see a candidate who has core beliefs and a modicum of loyalty.