Theo, yes you need a TC with the pz contacts to drive the SDM autofocus.
There is more than one version of the Kenko 1.5x pz. Not that the AF will be different, but the magnification is. The one pictured in your post looks the same as mine. Its real magnification is about 1.4x. (It is believed to be identical to the Tamron 1.4x pz.) It works well with my FA*300 f4.5 (effectively 420mm f6.3) in reasonable light; should be even better with the DA*300 because it's one-third of a stop faster (ie effectively 420mm f5.6). The other version (which is said to be identical to a Promaster Spectrum 1.7x) has a slightly smaller rear element (27mm v 25mm) which sticks out a little further; that version actually magnifies by a little more than 1.5x. Read more here:
https://www.dpreview.com/forums/thread/3537477
I find the TC hardly affects IQ at all. A comparison between a Tamron 1.4 pz teleconverter and the DA teleconverter in the PF review showed no practical difference in IQ:
https://www.pentaxforums.com/reviews/pentax-da-14x-aw-af-rear-converter/telec...omparison.html (Corner performance was a little better with the DA, but that doesn't usually matter for wildlife shots.) The main limitations of third party models compared to the DA TC is that they are not WR and they don't pass focal length information to the camera (so the EXIF shows the wrong FL and SR is affected).
The $64 question with teleconverters is: at what point do you get better results with the TC than you would get by using the lens without TC and cropping?
That will depend on the sensor of course. With a 24mp sensor and a sharp lens, there is a lot of scope for cropping - especially if your images are mainly for viewing on screen rather than for large-size printing.
My Kenko was about $200 second hand. If you haven't already upgraded the body from K-50 to K-70 or K-3, personally I'd say there was more value in putting the $200 or so towards that first. If you get a K-3 you will also get a big improvement in AF performance with every lens.
Edit: I note that PF member Klaus, who contributed to the DP Review linked above, reports that the version of the Kenko 1.5x with the 25mm diameter rear element (that is, apparently the same as the Promaster Spectrum 1.7x) does not work properly with SDM lenses. As for the other version, see Audiobomber's post below.