Why have a copy of a British gun when you could have a British gun?
$4K will buy a nice one in 12GA.
Some really nice ones for that money or less, for sure. The only issue is that most of them will be 2 1/2" guns. If that's not a concern for you, and if you don't belong to a club where they require steel shot or do a lot of hunting where you need to shoot steel, that's an excellent way to go. Of course you could spend $3K or less on a nice Brit boxlock and have enough left over to buy something that will handle steel, if that's only a minor issue for you.
