6.5 Carcano
6.5 Arisaka
6.5 Mannlicher
with pointy bullets 120-130 grains (7.8-8.4 grams).
Indeed, the different 6.5 mm cartridges make a lot of sense here, especially when looking back from today's era.
The 6.5 Swedish was also a Norwegian cartridge, and the 6.5mm was both Austrian and Dutch cartridge, so either will be easily available in Germany of late 1941. All of the European 6.5mm types will indeed need the spitzer bullet to work well.
BTW - Wkipedia does not note that there was a spitzer bullet for the Norwegian 6.5mm, but the German data plates do (9g bullet, velocity V25 of 745 m/s ) - ie. a bit weaker than the Swedish spitzer, but here this can be an advantage.
I wouldn't worry too much about a few mm of difference, after all the FG 42 worked with the long German cartridge.
I was trying to find the merit for the more powerful cartidges for this task, such are the 7.5mm French or the 7.65 Belgian (both being very modern for the day), but choosing these offers just minor saving in the forces for the weapon to withstand, so Germans might as well and go with the historical bullet.
Another cartridge that was pretty 'mild' when compared with the 7.92x57 was the Italian 7.63x51. With just under 2400 J, it is under the Arisaka's ~2670, let alone the big ~4000J of the 7.92. Italians were under pressure to miliatrize fast, and the 7.35mm stuff just didn't had enough of wherewithal to replace the 6.5mm, so they might be grateful for a shipment of, perhaps, French infantry weapons & ammo to the units in N.Africa in exchange for shipping the 7.35mm ammo & barrels to Germany. When compared with some 6.5mm cartridges, it is already with the spitzer bullet.
Yes, the 6.5mm cartridges with spitzer bullets will outperform it as the distances increse.
FWIW,
here is Gun Jesus, and hi's opinion about the 7.35x51 might surprise you