Bloomberg wouldn't necessarily run as a Republican. More likely: third party or independent.
If all we're talking about is Murdoch tweeting from his yacht on the other side of the planet, maybe Bloomberg won't run at all.
I doubt Bloomberg will run, he doesn’t have much of a fan base outside of New York City, especially among Republicans. The bigger question is what the RNC will do to guarantee that an establishment figure gets the nomination as Jeb’s numbers continue to stagnate. Will they flail around looking for a last-minute replacement (as Democrats may have to do for Hillary if she’s indicted), or will they just find a way to rig things so that delegates vote for Jeb or a similarly acceptable candidate if things don’t go their way.