So after way too much wrangling.. we went with the Century Southern Tower. It's a known quantity, rooms are reasonably big (26+m2 for a double), Y16,000 for their 45% off deal was in the ballpark, and while it's not physically 'near' where we wanted to be, it's located literally right next to Shinjuku station with stupendous shopping/dining nearby and we can very easily get from there to Narita (NEX), Tokyo station/Chiba (Chuo Rapid) etc.

Nevertheless, here's a few other interesting places I found in my wanderings. Maybe next time...
* JAL City Yotsuya actually looks pretty good. 20m2 rooms from Y16,000, and that can be paid in JAL miles.
* City Pension Zem is a quirky little family-run apartment hotel in Ningyocho, right next to TCAT... and not much else. 31m2 from Y13,300.
* Citadines Apart'Hotel Shinjuku does one-night apartment(-ish) stays from Y11,000, with microwave, laundrette downstairs, etc,. Rooms look pretty small, and it's two subway stops out from Shinjuku proper though on the wrong side of Kabukicho.

Now to figure out the onsen and we'll be all set!