I have yet to have a bad experience with tokyo hotels. You can't go wrong with the choices listed.

I stayed at the princes gallery right before the pandemic started. Good breakfast with i believe ability to order from the menu in addition to extensive buffet. If I recall, I got the corner room and it definitely felt bigger than the andaz or PHT room. The aesthetic overall and the bathroom in particular was fine but not remarkable. The lack of club access is unfortunate, especially if you plan to host guests in the room. location was much more convenient than andaz or PHT, but i'm sure that could depend on where one needs to be.

It's been many years since i stayed at both the andaz and the park hyatt. Agree with all the comments here about the PHT. luxurious but dated. I was not a fan of the breakfast buffet at all, especially when compared to andaz or princes gallery. Room was fine but had a very old school aesthetic, not the clean decor of most park hyatts. One of the best hotel showers I have ever experienced. NY bar was fun but i wouldn't pick the PHT for 5 nights based on going to the bar one night. Location is a bit of a walk from shinjuku but not unmanageable unless you have a ton of luggage or you're there in the dead of winter or heat of summer.

I still love the Andaz. Great aesthetic. Excellent buffet breakfast. great bathroom. good gym. Location is a bit annoying. I found it not as convenient to public transport as the other two hotels. The rooms are smaller than the princes gallery and limited chance for an upgrade beyond better view. Also i don't believe the bathroom is as private which would make hosting guests a bit tricky.

I don't know anything about the edition other than there are limited (or no) meaningful status benefits offered, but that shouldn't be an issue if you do FHR or virtuoso booking.

Overall, i'd probably still pick the andaz with the exception of having to host people in the room. If it has to be in the room, i might go princes gallery, especially if you can get club access for hosting instead of your room...although be careful they may not let guests into the club.

One last thing... i had good concierge experiences with both the PHT and Andaz. I didn't try to use the concierge at the princes gallery.