Well, it's a Westley Richards rook rifle - a nice one of the type built on a Martini action. I see this one is good out to 400 meters (WR always seemed to psyche their customers up by putting really tall rear sights on their rifles.)
My books show the Sherwood with a case length of 1.550. Base diameter (above the rim) is .3201 tapering to .3181 at the neck. .3748 rim dia. So it's in the ball park and the Sherwood is a WR proprietary cartridge fielded to compete with the 310 Cadet. How long after the cast did you take the dimensions? I ask that because a .299 bore dia. and a .300 case dia. only leaves 1/1000th for case wall thickness. That's thin. Too thin. Even considering they made much thinner cases in those days. I haven't done a chamber cast in a long time but I recall that the cast takes and hour or so reach full dimensions.
.300 Rook is: OAL (loaded is 2.0402 Case length 1.152. Rim diameter .3748. Base diameter (above rim) is .3189 tapering to .3181 at the mouth.
I think it might be, looks like they cut a generous lead (throat). My first step would be to locate a loaded .300 and try that first before I bought anything else.
Look at this description from Christe's Acution house - if it is correct .300 Special and .300 Sherwood may be the same ? The Rifle described matches mine. My serial number is aproximately 5700 lower.