Brett:
You are highly unlikely to get a game at Tokyo Golf Club, unless you know a member, or possibly if Gil Hanse will introduce you to someone there. It's about as private as a member-owned club can get, outside the USA.
Kawana is the only one of the great Japanese courses that's easily accessible if you are not well connected. It's a bullet train ride and a van ride from Tokyo, but on the wrong side of the city from the airport. Not doable unless you stay overnight. Staying overnight to play Kawana would be worthwhile.
The one other thing I could suggest would be to play at Keiyo Country Club, which we have plans to rebuild one of these years. [The ownership has put off the project for a couple of years due to the downturn in Japan.] The club is on the north side of Tokyo, only about 30 minutes from the airport; and though it's private, it's not exclusively private, and I could probably arrange for you to play there. It's only an above-average course, but it might make for an interesting before and after comparison.