Congratulations! Unfortunately you have left us (and the Wisconsin Tourist Federation) wondering why it ran any better or more consistently when set up for HV mode!!
There has to be a clue in there somewhere related to the (apparent) fact that it started consistently when bumped with either one of the LV winding sets connected, but sat and vibrated with both connected. I will not be able to sleep tonight till somebody figures that one out. (Or until my bedtime, whichever comes first.)
First, I was testing it unloaded. Granted a belt-driven fan is not a lot of starting load, but it is some drag.
Second, when I hooked both winding up, unloaded, it did start. That seems to have been a "Yea, all three this time just to screw with him..." move by the breaker. A subsequent attempt also started but in single-phase mode.
That still does not give me a clue why it worked consistently with the series setup, unless somehow the extra inductance or the lower current affected the behavior of the defective breaker itself.
My Guess: Crap shoot; the breaker excited all three phases that time.