You are right at the point where I was my first time with the GPS unit on the K5. I had everything setup just like you have and I kept pressing the shutter (it seemed logical to me). I finally went back and read the GPS manual and saw that I needed to select the "start shooting" option with the [ok] button which flipped the screen to a display that indicated that the camera was ready to track in astrotracking mode and then hit the shutter - and then it all worked.
In your case with the K-S2, they have eliminated the "start shooting" option, and substituted the - "press [menu] button 3 times" and then press the shutter (page 118 step 6 in the K-S2 manual). I have no idea as to why this approach - but then again, I didn't write the software (user interface). It sounds like you need to press [menu] button 3 times followed by pressing the shutter, to get the camera body to enable the sensor tracking capability, when the shutter is released. In the manual I see no instruction to press the [green] button.
[just an aside - I would boost the ISO to at least 800, if not 1600 to 3200, in order to capture enough star light.]