you should be able to do this through trial and error, by changing the track a few cents at a time (if it's between keys)
I'd undo each failed attempt and start over from the original track with a new transpose value. That way you won't have multiple transposes on the same track, which would probably degrade the quality more than one transpose.

Another possibility: If you have even the most basic version of Melodyne, it should do the job nicely.