If it being a traditional style weapon is not important, have you looked at the Traditions Pursuit Shotguns?
I have one in 12 gauge. Traditions Pursuit Pro. Chrome lined barrel, 209, break action. And changeable chokes. Mine came with Extra Full. Currently it has an IC choke in it. Works well for turkey (Extra Full or Full), rabbits, squirrel with the IC choke. It has fiber optic sites.
It cost about $400 a couple years back. Price and function are both nice. I'm sure there are better ones out there, mine is good for me. I have shot #4, #5 and #6 from it. Varying charges of loose 777, FFFG, yeah I know FF is the correct granulation. I use what I can get.
Haven't tried it on birds in flight yet. The stock is not as straight as many point and shoot guns. However it may be fine for pheasant etc.