Can you try a different battery?

I don''t believe attempting a direct firmware downgrade will work but have not needed to try it. It certainly didn't work with earlier Pentaxs. You'd think there would be an easy method to do this though.
I believe the only way would be to modify the old version's code in the bin file to give it a new version number that is higher than the current one. Then to go back to the current one you'd have to modify that to a higher version number still. All at your own risk of course! FWIW I've not noticed the issue myself nor heard anyone else make mention of it.
If you have specific examples why not take it up with Ricoh? Maybe you've spotted something no one else has.

Edit:
I've just looked at the bin file in a hex editor and the location of the version number didn't leap out at me.

No, it's because it isn't!

Well presumed Mike! Tried the lens on older Pentax's?
If so, perhaps Mike could suggest why it would work on some bodies but not the K-1. The only issue close to that I'm aware of is a failure where the mount contacts open circuit so doesn't communicate with the main part of the lens. I've got one that's intermittently like that, but it doesn't turn the camera off. Something like this: https://www.pentaxforums.com/forums/58-troubleshooting-beginner-help/239946-sigm...

Which camera, which exact lens? Does it work for a time before it stops?
You say have also looked for the firmware, but I couldn't find any link so how do you know if its the latest?
A little more detail would be helpful to help you resolve this.

The 70-200 image doesn't show exif so no focal length presented. It still made the point though!

If it is intended to be a retro flagship, why isn't it being marketed as one?

The easiest way (for meat least) is to get the camera to do it itself at the time of exposure using either the O-GPS1 or the onboard one (K3ii or K-1). I find Image Sync unusable as I don't get an image through it! In any event I believe the app only acts as a remote control.
As to adding geotags in post processing I've used GeoSetter (freeware) with some success http://www.geosetter.de/en/falsche-url-zur-karte-in-version-3-4-16/
I've come across other programs that match exif time stamps with GPS route data to create geotags, but not actually used them for that purpose.
It seems you have had an alternative suggestion to this query on another Pentax forum so your choice!

Indeed really good. I like the way you've got something sharp to contrast the movement in many of them.

Google tells me that it is Leeds Bradford Airport.