Most Kar88s were NOT "upgraded to "s" specifications (.323 Bullets) and still used the M88 clip and cartridges in WW I; although some were "reamed out" to use the "S" cartridge (.323 Ball) , still loaded in the M88 packet clips.
NO knwn M88/05 conversions of the Kar88 are known to exist (Stripper clip models, with "S" tube.
That behind us, I would strongly suggest you load and fire correct M88 cartridges, or the Cast bullet equivalent of them.
NORMA used to make "8x57J" round nosed hunting ammo of the correct specs ( .318 Bullet, for .321 grooves); RWS still supplies it also ( 196 grain SP RN).
For cast Bullet loads, use a .320-.321 cast ( such as the Winchester .32 Special or the .32/40 cast bullets) to get a good gas seal, and correct barllistics.
American Commercial "8mm Mauser"....the Bullet diameter is .320-.321, the Loads are "down" for Litigation reasons lost in the mists of time, and I suppose, one "Could" use them for occasional shooting.
But remeber, Kar88s are from at least 1907 or earlier ( 1907s are commercial ones, Military are pre-1900)...so, have care for the aged ladies.
I would go the Handload way, keeping loads moderate, and using a good design .321 Gas Checked cast bullet, of about 200 grains ( round nosed preferred.)