"Kinda like the GI's putting 7.7 Jap in their Garand's if they were out of ammo or in a dire situation. It would fire, there would be a lot of pressure, but it's better than no ammo at all."
Pressure would actually be low if the 7.7 round would chamber in the Garand. The Arisaka cartridge is slightly larger at the case head, so complete chambering can be iffy.
If the round does chamber, the bullet is .311, as opposed to the .308 diameter of the American round. .003 isn't that much.
What would really work to keep pressures low, though, is the fact that the Arisaka case is 5mm shorter than the .30-06 and the case at the shoulder is significantly smaller -- .429 for the Arisaka and .441 for the .30-06.
That extra space acts as an expansion chamber, helping keep peak chamber pressure low.