Hi there, Grandma Bear,
When I tried to turn the name 'Bregman' into a, hopefully, pretty close German original, the following associations sprung to my mind. Your suggestion that he might originally have been called 'Bergman' is certainly a possibility. It this is the case, however, then you're probably in trouble, because I think there might be quite a lot of Bergman(n)s originating from Germany. 'Ein Bergmann' in German is a name referring to a profession: it means 'a miner'. However, I was also thinking that the father, John Bregman, might originally have been called Johann(es) Brechmann. As English people usually have great difficulty in pronouncing the German 'ch' correctly, and often wrongly pronounce it as a 'g' (as pronounced in the word 'girl'), the father might have deliberately changed the difficult 'ch' into an easier pronounceable 'g'.
If the name Bregman is the original name, however, then I think the 'e' in Breg would be pronounced as 'Brayg' in German, and not as 'Bregg'. Thinking further along this line, I thought 'Breg' could be a reference to the Bregenzer Wald. The father's ancestors could have originated from the Bregenzer Wald (the Forest of Bregenz, on the border of Germany and Austria), and because of that have been called Bregmann: a man from Bregenz. But this is merely a rough guess on my part. Perhaps the latter suggestion is absolute nonsense. Or is there a town or area called 'Breg' in Germany and were they simply named after this town or area'?
I haven't done any research into the possibility of the existence of these surnames. However, I have never heard of the name Bregman, but that doesn't mean it doesn't exist. The surnames Bergmann and Brechmann do exist, however, and there are people called Johannes Brechmann.
Hope this will help you.