Your earlier post was a bit ambiguous!

I'd expect both the camera and the lens to have to be on AF for AF to function. That's how it normally works.

One comment on the AF was interesting, "The AF system is faster when the focus ring is on AF."
Surely if it wasn't set to AF it wouldn't autofocus at all. Mmmm!.

Is it this one: https://www.pentaxforums.com/userreviews/sigma-70-200mm-f2-8-apo-ex.html ?
If so it doesn't appear too bad from the reviews there.
Best way is to try it as even the best lenses can have batch variations and that one looks pretty heavy which might put some off. If it works for you.... !

There are Duracells... and there are Duracells. Some good, some bad!

https://thecounterfeitreport.com/product/106/Coppertop-AA---AAA-Batteries.html

http://www.thedigitalphotoguy.com/2016/09/09/fake-duracell-aa-aaa-batteries/

Then the camera's off. Was a common problem with the K-5 though not so much with the K-3. Surprised you can't notice it.

Now you've let out Gareth's secret.

The HDR'ing is just fine, some photos just need it. Watch the horizons as they're all a bit off (not level!)

Hi, welcome back
This might give a clue (perhaps.)
https://www.freshnessmag.com/2013/08/01/pentax-ricoh-gr-digital-camera-customiza...
There is one similar shown.

Smart people. Until you've tried both you won't appreciate the difference.

Getting a new camera, a K-x or something else might seem a sensible option rather than possibly throwing good money after bad, even if just a tenner. But hadn't the OP reported it as being OK now?

To me it would make sense to try the update and if it fails replace the camera. Can't see there's much to lose. If it were me, though, I may be tempted by the DIY battery adapter solution. If the OP has lost confidence in the K-x getting something newer would make a lot of sense in any event.