Interesting. Bloomberg getting into the race is one thing that could give us a GOP president other than Donald even if Donald is the GOP candidate.
Since Bloomberg is so liberal, he might actually have a shot at winning a few northeastern states - not enough to win the presidency, but enough to prevent either Hillary or Donald (if he is the nominee) from getting to 270 electoral votes. If that happens the election goes to the House, where each state delegation gets 1 vote, with an absolute majority (26 votes) required to elect the president.
Now there are something like 36 states that have majority GOP delegations, about 12 states with majority Dem delegations, and the rest are tied. So you would think the GOP candidate would win easily. But if Trump is the nominee, I can see a scenario where enough state delegations vote for Bloomberg to keep Trump from getting to 26 votes, so that no candidate gets a majority. If that happens, the person who was elected vice-president (assuming their election is confirmed by the Senate) would become president.
So in that scenario, if Donald was the nominee whomever he had selected as VP could become president instead.
Hmmm... depending upon who the VP nominee was, it might be enough to make me vote for Bloomberg...
Bloomberg has the personality of a dead fish.
Bloomberg could never come close to winning in any state, including New York. Not against Trump, not against Hillary, not even if Sanders is the Dem. Trump could win those states as a third party, perhaps. Not Bloomberg. So your scenario is not plausible.