I don't think it's a balance issue? Being out of balance may make it shiver and shake a bit if it's really out of balance but at 60-70 mph, if it weren't balanced, you problably wouldn't notice much. And, if it's not spinning it's not loose.
If I had to guess, I'd guess it was out of alignment.
check the marks on the swingarm and be sure the axle is aligned on both sides. Being out of alignment would make the rear want to step out in one direction and stand up in the other direction.
If wheel alignment is not the issue, I'd look closely at the rider to make sure it's not his imagination
If all else fails, try dropping the pressure down to 30-32 lbs so the tire flattens out a bit. If it still does the nasty then it's not the profile.
Like BA Buse said, that is a cheap tire and it could be shape of the profile that makes it feel different from the original tire. Especially with a different brand tire on the front.