We stayed at the Prince Sakura several years ago and found it very convenient for the Circle line and for the Bullet Trains - we actually left from there to Kyoto/Osaka. Note it's also a very short train ride from Haneda if your flight arrives there rather than Narita. We had one of the large upgraded corner, "rounded" rooms (don't recall what they're called on the reservation system). There is no club so had breakfast in the restaurant every day which was extensive and certainly many veggie options - my partner is vegetarian. (ON EDIT: apparently there is a club per the sticky, so either it was closed or just not impressive while we were there!)

You really don't need to bring food back to your room just to stay vegetarian. The restaurant where they serve breakfast was an italian restaurant at night when we were there so obviously plenty of veggie choices. The grounds are lovely, plus you also have access to the restaurants and other facilities of the two other Prince hotels on the grounds, both of which are larger "conference" type hotels. The Sakura has a more boutique feeling to it. Could probably stand a refresh although to be fair the 70's decor is now back in style as retro. It's also nice that you're off the main road and in a more residential area so pretty quiet at night time.

The gym in the basement is actually a membership club so pretty well-equipped if that matters. We didn't use it but there is a pool which a 6-year old would probably enjoy.

Have not stayed at the Westin so cannot do a comparison