Well, "Run With The FOX", I believe I've probably seen at least a half-dozen Parker "Invinceables" with the serial number 200,001; and I suspect anybody who has been collecting very many years has also. I knew the gun was a fake when I picked it up; but overall, it was one of the better examples I had seen and quite attractive. As a matter of fact, it would be the perfect gun for your future bird shooting pursuits; you'd enjoy looking at it while waiting on that point, and any bird you shot would die much happier knowing that its demise had at least been effected with a beautiful gun! And further, there's the additional perk of having such a beautiful gun highlighting your bird and dog pictures afterwards; but the best part, all this happiness can be obtained for the paltry sum of $12.5K (chump change for the snow birds we regularly see down here)!
Seriously, I collected and shot only Parkers until my eyes were opened and I finally saw the light; but in the interim, I shot a lot of targets and game with many Parkers that also included a very late 16-bore Trojen that Del Grego had converted and re-stocked to straight grip. I actually wish I still had that gun; but fine guns are like beautiful women (and about as expensive!), and my fickle eyes were always wandering and fantasizing about my next acquisition.
As far as Trojen guns with a Parker single trigger, I've not personally seen one and don't recall such option being cataloged; but have heard from reliable individuals that a handful do exit (maybe Eightbore will chime in).
As to Trojen up-grades, there are apparently a hand full of those out there also. A few years ago, while this dumb hill-billy was wasting his time at an Atlanta gun show, a New York engraver had such a gun on display at his table exhibiting his skills; and although I don't recall the engraving pattern exactly, I think it was engraved in the Invinceable style!
And hey, I ain't proud; I'll drink with anybody, including Yankees, as long as what I'm drinking is no stronger than Coca Cola! To do otherwise is against my religion.