The 7.7 Jap will have a barrel diameter closer to .311 or .312 I think, so the .308 probably wouldn't work too well. Plus I don't think they make any chamber inserts, so unless you can find one, I believe you would have to rebarrel it.
And then I think you would have to do some work to the bolt-face too.
Changing the barrel and/or the original chambering is going to cost you MEGA bucks, so while it is fun to dream about it, the results are usually not worth the time, effort and money UNLESS you are going to go full bore/damn the torpedoes/money is no object/dammit I WANT a "one-of-a-kind!" But even then you will probably not EVER end up with a rifle worth more than the sum of it's parts.
Why not leave it as is, shoot it occasionally with the $17/20 cheap reloads from a gun show, or the surplus stuff you can occasionally stumble on (the 7.7 is easier to find than the 6.5, I know that much!) and go spend $75 on an decent old Russian M44 and get all the blasting ammo you want for about $35/440 rounds, and decent S&B or Winchester factory ammo for like $10 or $12 a box for hunting? Plus the Mosin's crappy safety is almost, but not quite, as useless and hard to operate as the Arisaka's.
On the other hand, if you still like to dream, remember that Old Arisaka action is the absolute STRONGEST B/A action ever made, so it just might be a cool starting point for a conversion...(IF you don't mind the CRAPPIEST safety ever designed...)