The engraving pattern suggests an early No. 4, and it has replacement barrels. After about 1898, Nitro Steel was available for the No. 4 so it likely started with Damascus barrels.
Later style engraving with the oval forward
You can find the date of manufacture here using the SN
http://www.lcsmith.org/shotguns/manufacture.html
And LOTS of information here
http://www.lcsmith.org/index.html
A Research Letter might show the family to be the purchasers, and if the HOT was originally ordered. A few guns 1901-1903 were fitted with the HOT, prior to it being introduced in 1904.
http://www.lcsmith.org/shotguns/researchdivision.html
Still an interesting and desirable gun, esp. if it can be connected to a turn-of-the-century "Top Gun" at the Traps.