I'll second the recommendation for Michel Troisgros at the Hyatt Regency, if you're in the mood for high-level modern French. I think lunch is quite a bit cheaper than dinner. And I would pick Kozue (the Japanese restaurant in the Park Hyatt) over Kakiden.

Tokyo doesn't really have a lot of options for shojin-ryori. Daigo is probably the best and the easiest to get to, even though it's not very close to Shinjuku.

Kyubei in the Keio Plaza might be your best bet for high-end sushi in that neighborhood; I can't think of any alternatives offhand. I've also heard good things about the Asahi Sushi on the 53rd floor of Opera City, which is pretty close to your hotel (http://r.gnavi.co.jp/g264803/ ).