I know the date of birth of his wife, I know the name of all or almost all their children and all the different places they were born, I know that he died sometime between their youngest child's birth in 1883 and my great grandfather's marriage in 1897. I am hoping that in the next few days the church where his wife was baptized in Budapest will be able to give me their marriage date (assuming that they married in that church) and at that point also, his date of birth and the name of his parents as well as confirmation that he was born in Fehring. Given that he worked and lived on cargo boats, it could well be that his father did the same, therefore it is not a given that his family was from Fehring, though there are and were some Baumgärtners in Fehring around that time. It is a pretty common surname in Hungary as well.
I only know that he was apparently born in Fehring because it is mentioned in one of the birth records for his children. The record for the first known child is recorded as Paumgärtner which makes me think that his mother tongue was German and not Hungarian and to non-Germans a "B" sometimes sounds more like a "P" when pronounced by a German-speaker.
I'm pretty certain that I will never be able to find out where and when he died because of his work. He will have been buried where ever he took his last breath, which could be anywhere along the Danube or one of its tributaries.
I have thought about contacting the Archives in Graz but I'd rather wait until I am certain of his birth date and his religion for that matter. His wife was RC, his children (male and female) were RC but that does not guarantee that he was.