I've owned 60 or 70 side by side shotguns but now am down to to less than ten.
My first nice SxS was a Merkel 47E with 28" barrels which I still own. It a 12ga but weighs 6lb 12oz. I've killed Canada Geese with this light shotgun. I much prefer heavier, longer barreled guns for waterfowl.
I like 16 gauges for upland but that's me. I want a Merkel 16/20 which are built on small frames. I want on of those by really 47E is light and handles well.
$3k will get you a decent SxS.
Are you looking for a 6 pound gun or one closer to 7?
Screw in chokes or not?
Do you want modern manufacturing or a vintage gun?
I like the AH Fox to include Sterlingworth in vintage American guns. For English guns, there are many. $3k will get you a nice 12 bore English SxS. The small bores are of course less common and more expensive. I do like the fit and feel of English guns.
There are modern 20 gauges mane by Huglu in Turkey that are okay if finished well.
The CZ Bobwhite may be a decent gun. Here's one for $650.
Buy CZ Bobwhite Case Colored Blued 20ga 3in Side by Side Shotgun - 28in at Sportsmans Warehouse online and in-store has everything for your outdoor sports adventure needs. Fishing, rods & reels, camping gear, tents and much more.
Probably not magic at that price point but maybe okay.