Assisted with input from a friend, I can answer my own question. Wills and admons for Snook of Stalbridge (and the surrounding district) are on ancestry. 

There is one admon and one will on ancestry that might relate to the correct Thomas:

 - Thomas Snooke admon 24 July 1683. He was a button maker with wife
   Melior. 
 - Thomas Snooke the Elder, will 1660. He was a Tallow Chandler
   with a wife Katharine. They had a son Thomas (who was left all the
   tools of the business).

The "K" on the farthing suggests Thomas's wife's name began with a K, so Thomas Snooke the Elder is the likely issuer of the farthing.

The "fleur de lys" on the farthing is likely to be one of:

- an emblem of the guild to which Thomas belonged
- an emblem of the trade
- a sign of where the business was located, or a nearby tavern.