If you want to use Hilton points, then the ideal place to stay is the Hilton in Shinjuku, if it is available on the dates you need. Well worth the difference in points, I would guess.

Laplap, I definitely wouldn't consider nishi-Shinjuku go-chome station close at all, in fact on the Oedo line, Tochomae station is closer to that hotel, there's an entrance on the other side of the park. And I honestly can't imagine any path from the Park Hyatt to the Hilton that passes near this hotel. But it's not relevant since the OP isn't interested anyhow. I wander thru this area A LOT when I am in Tokyo!