the short action and the curve of the loading "ramp" are obstacles working against you.  Its a mater of geometry, there is a finite limit to the length of case or cartridge you can slide down and into the chamber.   fixed ammunition is longer than if you were breech-seating a bullet and then inserting a properly prepared cartridge case.

it is true that incredible results have been obtained with the 32-20 case as well as cases based on the 357/mag/max case.     BUT most use longer heavier bullets and relatively high pressure loads----and I suspect most have been shot as breech eyed loads rather than fixed.  and, as far as I know in different, perhaps stronger, actioned rifles.

    I believe that loading the long heavy bullet as a fixed round  MIGHT work, but maybe not--depending on length. and then there is the matter of safe pressure limits  for the Cadet action.

I don't have any answers for you in your search.  I do have a number of "martini actioned" rifles  but all the centerfire ones are in larger european 
schuetzen rifles and most are chambered for moderate pressure target loads.  and frankly breech seating in them is a pain.   the European shooters did not usually shoot that way so it was not a problem for them.

Why not use 310? I have shot mine at 300yds at Bisley and it is still very accurate at that range.
Fred

Would a 31-222r (.312") or 33-222r (.338") be likely to be good rounds?

Hmmm.  My most accurate Cadet is in 221 but I haven't tried cast yet.  (Love the Bob Snapp extractor).  I'm having a 17-221 worked up now.

I had one in 30-30 - a bit too much to get in a loaded round, so it is now re-barreled to 7mm Int R.  (full length 30-30 necked down and sharper shoulder).   

Most fun is the 30-30 WESSON (straight tapered .357Max case).  30-20 or 32-20 would be excellent.