Henry R. Braunsdorf died in 1912, I suspect the company veered off after that.
Obit gives some hint of founding date...
" Mr. Braunsdorf began his business career as a stock boy with Hammacher, Schlemmer & Company, of New York, in 1882...
... In 1898 Mr. Braunsdorf associated himself with Charles F. Mueller, who was then conducting a tool manufacturing business under the name of the Charles F. Mueller Company. Mr. Braunsdorf became at that time one of the firm and later the business was incorporated under its present title.
...
For ten years Mr. Braunsdorf had been president of the Braunsdorf-Mueller Company, but for the past two years, owing to illness, he was unable to take an active part in the affairs of the company
(Building Age, Vol 34, 1912)