Since you asked... if you study the .303 case you will find that it is very much like the .30/40 Krag case and the .405 WCF is the same case a bit longer.
There was a .35 WCF on this case but it went no where fast. Awful close to the .35/303 you say you do not want. .303British.com can put you in touch with the Epps gun shop where Mr. Epps did all sorts of .303 mods. In hands of offspring, I believe. The other common, the .35 Remington is rimless. There was a pistol version using the .303 case (for the rim) and cut off a bit to get a rimmed .35 Remington. You load with .35 remington dies and a .303 shell holder. Obviously, .405 is an option.
Mauser and the other rimless used a bit larger head to achieve rimless. .356 is the Mauser base/head with a rim. Little big. Should work o.k. in the Martini... I would quiz the 'smith closely about enough metal left in action/barrel for strength. Age of gun???
As said, the militarys are getting less common. You might be ahead to sell the thing and buy something more modern in a caliber you like. eabco.com has a sweet little falling block in both rifle and pistol. He beats the drum for the .219 Don Wasp round. (and .30/30 Imp). Highly efficient. Luck. Happy trails.