Over that past 2 years, I've stayed at Andaz 4x, HR 3x, GH 1x, PH X1. My kids are now 8 &9. Although everything is very accessible by metro, by 6pm my kids are exhausted and are begging not to take the subway.
If traveling alone, I prefer Andaz. As a family, our go to place now is HR for the convenience mentioned above. With the miso,rice, and condiments added in the lounge since last year, it really elevated the lounge experience. They also have the best cappuccino maker I have had in any lounge. I can't stress enough how nice it is to have the government building with an attached walkway where you can go for delicious cheap (500-600 yen) lunch and, I believe, dinner when you are tired and jetlagged. There is a great variety there so you can try something different every time. My kids also prefer to grab something from Lawson's or 7Eleven rather than catch the subway. I agree that even with an unlimited metro pass, the shuttle is a very nice way to get to shinjuku station.
We also like to hit takashimaya/tokyu hand/basement food hall after a visit to shinjuku gyoen park since it's a short walk away.
Again, no "sneaking" will be necessary with 2 toddlers. Do make them aware ahead of time and hope they will put you in a larger room.
Lastly, using C&P, b/c of fees, its less than 1/2 the
price to stay at HR as it is at Andaz.