Beretta 390 for under $750 used. Spend the rest on ammo and professional instruction. Spending the high limit of your budget will not get you a better gun and it won't make you hit more clays.
The one that fits.
And mostly THIS
Remington 1100, Winchesters, Beretta 390, 3901, 391, 302,3,4 and many others all have taken their share of shooting trophies at clays. Find a good instructor in your area and take lessons. The first one should be to check the fit of your shotgun.
The Beretta's have shims that will help the shotgun fit you better.