How about a light English boxlock 2" or 2.5" 12 gauge. many of them are 6 lbs and some less...
There are also 16 and 20 gauge Fox Sterlingworths that weigh 6 lbs and are priced well under $2k.
The problem with the 2" guns is ammo. If you're only going to hunt with the gun and not shoot many targets, no problem. But it gets expensive if you shoot a lot. A few people do reload 2", but not many. 28ga, on the other hand . . . easy to reload.
You'll probably look long and hard for a 2 1/2" Brit gun that's sub-6#, and the ones you do find will likely be 25-26". But if a little over 6, up to 6 1/4 is OK, there are quite a few more out there. My Lancaster SLE, 28", is just a bit under 6 1/4.
