As a long time resident of Tokyo and someone who practically lived at Grand Hyatt and Andaz, I think you’re making a huge mistake scratching those off the list. Conrad is also a far nicer hotel and location than Shinagawa.

Your point about airport transportation is valid, but I don’t think it makes sense to stay in an otherwise inferior hotel/location on that basis. A couple of inexpensive taxi rides would completely solve your problem and leave you staying in a much better hotel.

You do realize that, for example, Shibuya is a very short taxi or bus ride from Roppongi Hills, right? And that a taxi to Shinagawa would also be easy to take from GHT, Andaz, or Conrad. Also, N’EX train to NRT runs out of Shibuya, or there is limousine bus service directly from all of those hotels? A taxi from Andaz or Conrad to Tokyo station is also cheap and easy.

Bottom line - take the best hotel you can find within budget that has availability. Then later figure out how to fill any transportation gaps.