1845/1846 listing so he must pre-date 1844 seeing it would take a couple years to obtain the dubbed title of HofBchsenmacher.
F. Berger may have been the father of Rudolf Berger:
"Der sehr renommirte Bchsenmacher Herr F. Berger aus Kthen hatte verschiedene Jagd- und besonders Vogelflinten von zum Theil neuer Construction und vortrefflicher Arbeit ausgelegt die vielfach versucht wurden und grossen Beifall fanden."
1856 listing
Vogelflinten reference is interesting.
There was a Joseph Berger of Gratz active in the early part of the 19th century. He received a bronze medal for his wares in 1841.
Kind Regards,
Raimey
rse