There was an LC Smith pigeon gun available a few years ago that had his name on the trigger guard but no provenance whatsoever. I had the opportunity to buy it, but I just couldn't make myself pull the trigger on the weight of the name alone.

I've seen a couple of what you're calling "medallion guns" and believe them to have been fake. One was a cheap single shot being offered by Sold USA with some kind of glass ball championship medallion on the stock. There was never a glass ball championship by the name that was on the medallion plus the dates were off from the actual glass ball shooting era. The other one I saw had problems as well, and both of course had absolutely no provenance.

I'm unaware of a gun that can conclusively be provenanced to Capt. Bogardus without question. But I've not seen the Greener that you say the ATA has, so maybe it straight.