I don't the Browning's would be the way to go as there is not much similar at all between them and a Glock.
The conversion is a good idea though. There are several companies, Ciener, Advantage Arms, etc. that offer them.
As for as a complete gun, there is a Glock copy imported and sold by "Austrian Sporting Arms" located in Ware, MA. I have handled one but never shot one, so I can't speak to reliability or accuracy. It is a very close copy of a Glock though but is smaller than a 34 (more like a 19 if I remember right). It has the trigger safety but also has a slide mounted decocker which of course Glocks don't have. It is marked "ISSC M22" on the slide and "ISSC Handels GmbH" and "Made is Austria" on the frame. Price was in the $350 range if I remember right. I think if you Google some of the terms above or "Glock .22 copy" you will be able to find some info on it.