You have done a ton of research, and it's quite interesting! I did know about your grandfather Harry (I am pretty sure we have pictures of him), I don't remember ever meeting him, I do remember my grandfathers' half sister, Betty Bohan, quite well (although I don't see her in your tree).
We were always told that the name change of 'Joseph Barry Bohan' (they called him 'Big Joe') was due to the fact he was trying to dodge the WWI draft. Note the discrepancies on the 1920 census vs the 1930 census in age and birth place.
The story we heard about Big Joe was that he came from a big Catholic family, and when he joined the Salvation Army, they disowned him. I am not sure if contacting the Salvation Army would shed any light. I think Big Joe had a lot of stories!
I always thought that they moved to Arizona for my Grandfathers sisters asthma, but after looking at the records, it didn't make any sense.
My Grandfathers' mother (Maud McChristy) died when he was around 12-14. At that time my Grandfather was left with the 'Perez' family in Kansas City and lived with them for quite awhile (a lot of that is murky). So interesting to note Big Joe may have done the same thing his Father did!
Did you find any relatives of the last remaining Bohan, Margaret?