It's not bad in the sun at all. I'm always shooting into the sun and it's fine. Of course it can flare, but not worse than anything else in my experience.
It is slow to focus, but it doesn't loose focus and hunt laboriously like the 300 does. I wouldn't want to chase something random like a child or dog, but predictable sports like motor racing are not much of a challenge for any lens really.

The K10D is a splendid thing, but you'll need lots of light as it is not the most sensitive thing.
Unless you pay big bucks you'll end up with a very dark lens that will be prone to camera shake and so not the advantage you might be hoping for.
I paired a Tamron 70-300 with my K10D for over a year and it was a very capable combination for very little money (about £60). Faster focusing than the 55-300 but a little prone to Purple fringing (easily fixed) and softness at 300. perfectly OK for web or small prints tho.

That's actually a fairly small range. If you have enough money for a 50-135 then you may be better served with a prime and use your feet to zoom.
I love my FA77 ltd and when it was working I never used the 50-135. That is essentially a wedding lens (for me), where you need the zoom for speed.
But for portraits it's no contest - the 77 wins hands-down. Better bokeh and 3D-like rendering that looks almost Photoshopped.

I think you need to link to the shot if you want us to zoom in
The 50-135 is very sharp wide open, with some PF predictably. The bokeh is slightly nervous on leafy, sparky backgrounds, but the contrast and colour is stunning. Very versatile.

Trying to imagine this from the inside. I think deep shadows would be normal and even desirable to emphasise in places.
In a rush I would try HDR or double exposure route with very careful and subtle bringing-up of dark areas back home in post. The K5 would certainly be able to do this with just one raw shot, not sure about the K3 tho as I don't have one. I can't imagine the pop-up flash working - at least not without lots of help to create depth and interest, in which case...
with more time, I would take an off-camera flash and try firing it in from the outside, creating lovely shadows and textures. You could take multiple exposures moving the flash and knit them together to make better use of it. Maybe even add some smoke or dust (subtle, not cheesy).
Footnote: I routinely take bracketed shots and routinely only use one of them when I discover the K5 has all the info I need

Ha ha, excellent stuff, love it
Looking forward to seeing the next one

It depends what your objective for the trip is of course.
I took off to Portugal for a week in a hire car with just a WP camera. Didn't miss my kit bag one bit. Very liberating.

You can stitch together multiple shots with the 18-135 but you can't zoom in any further than 135. So I think I would go long. I'm partial to long lenses for sunsets if you have the right landscape to get back.
I love traveling light and not changing lenses tho, so I would even consider not taking anything more the the 18-135 and just enjoying the freedom.
Maybe get right out of your comfort zone and restrict yourself to a prime!
I'm not helping at all am I?

Bump

Put the lead in the remote release cable port.
I don't have a K3 but it's down on the right, under the rubber flap I think.