The MM designations on a German gun can be very confusing. There were a plethora of cartridges that would fit in the category. 9.3/8.8/8.7 could cover a multitude of bore and groove combinations. My 9.3x72 has a .344 bore and .364 groove dimentions and I've heard of them going as low as .356 in the groves and still be a 9.3.
Mule Deer's wife had a drilling with a .358 groove diameter.
Here is a discussion on a 9.3x72 that sounds like yours.
http://castboolits.gunloads.com/archive/index.php/t-75108.htmlActually you are kind of lucky as the .358 groove diameter opens a whole world of bullet options to you that aren't available for .364 grooved barrels.
Keeping your loads in the mild range you might never need mor than a .35 Rem die set to do your loading and a 9.3x 72R FL sizer to size the body on occasion if you start with new bras.
If you can load a couple of new brass with 38 special or 35 Rem dies you can send the fired brass of to C&H and they will make a set of dies that will work perfect very reasonably.
C&H has a lot of experience with the odd sized 9.3's
If you think 9.3's are strange 6.5's in Euro arms run from .250 to .268
The 360 marked case you saw was for the British .360 Nitro Express 2.25", brass for it can be make from shortened 9.3x72R brass.