Take a look at the F.A.I.R. Iside, which is available for under $2,000 in the base model. They are very nice guns for the price point and are available in 16 and 20 gauge.

The Dickinsons are also nice guns.

V. Bernardelli sxs were nice guns. I have a Gamecock 20 gauge I've owned since 1977.

Besides the Ugartcheas, the AYA 4/53 boxlocks are quality Spanish guns, but probably somewhat above your price point.

Fox Sterlingworth and some Fox A grade 16 guns will be available within your budget if you're not looking for a collector level gun.