I think the point of this thread is to recommend a decent hotel to the OP, not to discuss whether Westin is a 5* or 4*, it doesn't matter really as long as it fulfill one's travel needs, right?

To the OP, Westin is in the Ebisu area, which has a lot of dining options but imo, not so much shopping...

Sheraton Miyako is a residential neighbourhood, so not what I would recommend to a tourist/leisure traveller.

Hyatt Regency is in the wrong side of Shinjuku, as it's in the business (Nishi-shinjuku), rather than the shopping and night life district of Shinjuku (Shinjuku-3-chome).

If I have to choose between the three, I would choose Hyatt, as I can still cross the Shinjuku station to Shinjuku-3-chome (15-20 min walk) and its surrounds for shopping, dining and nightlife.

However, since you mentioned a Hyatt, you might want to consider Grand Hyatt? It's located in Roppongi and is a fine decent hotel with plenty of entertainment and dining options in the surrounds.