PIRANHA
Greek Redneck
I have a Remington 1100 Express that I have used for bird hunting since I was about 12 and I still have it and still use it. It is an amazing semiautomatic 12 gauge. And super accurate.
I'm a Beretta fan, would look for a deal on an A400 variant might find one close to your $1k budget, it's as sleek and smooth handling as the Benelli inertia driven guns, but has the light recoil and reliablilty over a broad range of loads of a gas gun. Excellent craftsmanship, lightning fast action, and probably the most reliable gas system around, recoil with heavy loads is tamed greatly with a kick-off model. Obviously you should handle and shoot several and see what works for you.
The Stoeger is ALOT of gun for the money. Solid choice of you don't want to go Benelli. One other point, I really don't think there's a whole lot of benefit to the 3.5" guns. There useful capability over a 3" gun is pretty slim. Hell, in all honesty, I use 2 3/4" for nearly everything and don't feel disadvantaged at all. I don't hunt geese though. JMO