The more I look at the gun after blowing up the photos, I think Dave is correct. The birds on either side would be more likely in the 2000 serial range of A and B grades. Of course this would be an A grade and out of the Paragon serial range .. I think if it is in the 2000 serial range, the engraving is actually etching. It's hard to tell with the photos given.

Barrels look to be 2 blade damascus and the fore end checkering is consistent with early A and B. From what I can see of the engraving it matches early A and B grade engraving.

Baker
 

Griffin & Howe always puts the serial number in their inventory number. So, it's 2256. It's there at the top under the GI number. And it looks like it has the protrusion on the lug. But, you sure don't get very much info out of BB on these guns.

BAKER, W.H. & CO.
Previously manufactured in Lisle, NY, production was done in Syracuse, NY 1877-1883.
The company was started by William H. and Ellis L. Baker in 1877. Leroy H. and Lyman C. Smith financed the new company, W.H. Baker & Co. In 1880, L.C. Smith bought the interest from the two Baker partners and continued production with "L.C. Smith Maker of the Baker Gun" marked on the rib and "Baker Pat." on the locks. Smith decided to drop the Baker name in 1883 but continued to manufacture this gun and a three-barrel shotgun/rifle combination in Syracuse, NY. The company was sold to the Hunter Brothers in 1889. A new company, Baker Gun & Forging Co., began making both the New Baker shotguns (see separate listing) and the Ithaca gun. The Hunter Arms Company made L.C. Smith shotguns for about 55 years, until the Marlin Firearms Company bought the business in 1945 and formed the L.C. Smith Gun Company in 1946 which produced shotguns until 1950.
Baker guns were originally 10 or 12 ga., and unusual in that the opening mechanism was operated by pressing forward on the front trigger. While relatively uncommon, most original Baker guns (including the three-barrel shotgun/rifle) seldom have much original finish remaining. Most are priced in the $400-$800 range, assuming finish is less than 10%. If condition is better than 40%, guns should be evaluated individually for accurate pricing.

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Riddle is solved. Good work, and a rather nice A Grade.

Baker A grade
 

As typical with this forum, it's always interesting and educational. Now that it has been identified as an A grade, what would be a more realistic value for this gun?