I think you should be doing pretty good with
the K1ii, although
the K3iii would probably be the more obvious choice for wildlife if you were buying a new camera and had a large budget. You already have
an excellent lens for that purpose, probably the best one available new in the mount.
There is no current teleconverter to go with that lens and camera, the only current Pentax brand teleconverter is
the 1.4x that's for crop only (without vignetting). It does work with that lens, more or less, and
there's a very recent article that goes into detail on that topic.
There are
older third party teleconverters (that 1.4x/1.5x version as well as 1.7x and 2x versions) that will attach and should pass autofocus through, though they may or may not make things too dim for autofocus to work well. "PzAF" is a good search term to use if you're looking for them. They will not give the camera accurate focal length (with the new FL after the TC), so the camera (for shake reduction and EXIF data purposes) will think it's using just the lens at its original FL. This will reduce the effectiveness of SR, but not completely.
Do also remember that you can use your K1ii in crop mode, which will help fill the frame with the subject (aka 'give you more reach,' if that's how you think about it (although some people may argue over terminology there)), as well as speed up burst and transfer rates and reduce file sizes. It won't get you anything from an image quality standpoint that you couldn't get by cropping on your computer, though. My favorite way to achieve that setting is with the top function dial, which you set to "crop" and then turn the third control wheel while looking through the viewfinder until you see the bold lines of the smaller crop area superimposed on the focusing screen.