You are absolutely right to focus on Lviv/Lvov. The idea that there was a sizeable Jewish community in the tiny village of Lemberg doesn't stack up whereas there is a long long and at times sad history of Jews in Lviv
.
The family story that he was from Austria is understandable, at the time he was born and up to 1918 Lemberg/Lviv was in the Austro-Hungarian Empire, often imprecisely referred to as Austria. I had much the same thing with my paternal grandfather who came from what is now the sovereign state of Croatia but in his time was a part also of the Habsburg monarchy.
Enough with the history already! Good luck!
PS - Haven't heard of other people adding to one's own tree - don't like that idea at all.