I would be looking to travel light and take either the 16-85 or 18-135.

I find the 16-85 to be a bit of a sharper lens than the 18-135, and for me the extra 2mm at the wide end is quite useful.

But that said the 18-135 is still good lens and I wasn't unhappy with the output from that. The longer range of that lens may be more useful for you on this trip, and would quite likely cover most of the shots you take.

Combine one of those with one or two small and relatively fast primes and you should have enough for most situations.

The 50-135 is a great lens, but lugging it around for 2 months might get a bit tiring.

Have you turned off highlight and shadow corrections on the K-3. Having them turned on does slow things up.

Also, don't forget that the file sizes on the K-3 are larger and will take longer to process and write to the card.
Are you using fast enough cards in the K-3?

I like the images too, but have to agree that they look a bit over saturated on my screen too.

Comparing images 7 and 8 I prefer the composition and the more natural look of 8

p.s.

Personally I would look for a recognised brand - Energizer, Duracell, Hahnel etc. where their quality control is hopefully better. Wasabi batteries seem to have good reviews, although I have no experience of them.
I have had a 'no name' laptop battery before, which failed very quickly, followed by overheating and swelling of the battery. (Needless to say it got disposed of very quickly!) Since that experience I would rather pay a little more for a 'known' brand.

I use an Energizer battery (which has exactly the same specs as the Pentax battery).
I've had that for probably 3 years now and it's just fine.
It seems just as good as the Pentax battery. I've never been able to discern any difference between them.

pm sent re: limited case

I think for something like tennis you may be better off turning off SR completely (camera and lens) and trying faster shutter speeds of not less than 1/300th and maybe try 1/500th or faster just to see what sort of difference that makes to the sharpness.

Either shutter priority or TAV mode and let the ISO take care of itself should work pretty well too, rather than full manual mode. The ISO performance with Pentax is generally pretty good, so let it go up to 1600 and beyond if needed.

Another thing to try if you are following the action is using AFC mode with user defined (centre) focus point and Expanded Area AF. I find that works fairly well for most action shots. Once you have locked on with the centre point then it tracks the action reasonably well. I also find that using the back button AF makes focussing easier than the shutter half press (although it may take a bit of getting used to)

I had a Sigma 70-200 and found it to be a very sharp lens ideally suited to action shots.

One thought is the Shake Reduction. I had problems with that when panning before I realised that the SR doesn't really work for panning shots! Turning it off and using higher shutter speeds improved things straight away.

Have you got it turned on both on the lens and in the camera? That could be one cause of not very sharp images, as the two systems would be working against each other.

Are you waiting long enough for the Shake Reduction to stabilise?

For panning shots the Sigma lens (or at least the one I had) does have a panning position on the SR which seemed to work quite well for motor sport pictures.

Sites like this may help to give an idea of tax and duties to expect
https://www.dutycalculator.com/

Why not just convert it to screw drive only. I paid for a repair on one once, and then regretted spending the money and wished I'd just converted it myself. It's a great lens, but that SDM motor does seem to be a weakness! A little bit of googling will probably send you to pentaxforums.com and you'll find the instructions there