As Ranch13 said, get thyself a chamber casting!
If you don't have any cerrosafe, a very hard parrafin wax can do in a pinch, but it is far more fragile.
The Martini Cadet
"Chambered for the .310 Cadet cartridge (aka: .310 Greener), it was used from 1891 to 1955.
The rifles will often chamber the similarly sized .32-20 Winchester and fire with some accuracy.[2] However the 32/20 is actually 0.312 cal and the 310 is 0.323 cal. "
"Firing a 120 grain heeled lead projectile at 1350 ft/s the round is similar in performance to the .32-20 Winchester and some rifles may chamber both rounds with some accuracy."
You mentioned beagling your mold, but I am not sure if you can get it large enough.
If you can't get legit .323-.324 lead bullets, you might try a pure lead 9mm or .38
and rolling it between two steel plates until it squishes down to the desired size.
a a 10/32 drill bit is .3125 and a 21/64 drill bit is .328 so two of these could be used as spacers.... at least it could get you close!
oh, just FYI modern Lee 8mm Mauser molds throw a .323 174 gr bullet, if you
really want to cast a bullet then cut it down....
here are the dims for .310 Cadet
from
https://loaddata.com/Cartridge/310-Cadet-Martini/2341this one sounds just like yours
http://castboolits.gunloads.com/showthread.php?248822-More-Cadet-Talkand this might helps
https://www.milsurps.com/showthread.php?t=10326pf mvl