In such a scenario, a third party would emerge. Tea-Party Republicans would never support a person like Dewey, who was moderate-progressive governor in New York (social welfare programs, minimum wage, anti-discrimination laws).
This would give Hillary the White House: Hillary receives most of the liberal-progressive vote and some moderates; Dewey gets most of the moderate vote (maybe some moderate-conservatives too), with the Tea-Party only receiving the far-right vote.
Yes, I agree that Dewey would be unpopular with the Republican voting base today. I couldn't image him winning against Hilliary Clinton if it was a one-on-one general election.