Good Morning, I have both the 12-24 and the 16-45 and like them both very much. I do think that the 12-24 is a tad sharper than the 16-45. Maybe I have a great copy of the 12-24 and a soft copy of the 16-45 - I don't know, something I should probably look into, one of these days. Its been said that the 12-24 is like a set of primes. It is definitely sharper on the low side and softens on the high end of the FL range. That said, according to Photozone, where the 12-24 and 16-45 overlap, the 16-45 is sharper in the center.
I was going down a similar part as you are now. See
https://www.pentaxforums.com/forums/pentax-slr-lens-discussion/78232-20-21mm-...der-zeiss.html Through a happenstance, I wound up with a FA 31 that I use for stitching. It is extremely sharp, across both the center and boarders. The Sigma 30 is sharper in the center but much softer on the edges according to Photozone. The reason I touch on this is that stitching a couple of vertical images from the 31 is sharper than anything, any wide angle can produce. This is not too surprising, in that a couple of stitched 31mm panels (portrait with overlap) has about 50% more pixels (horizontal) than a wide angle lens. It is all what you are going after, and how you use the lens.
There are countless threads comparing the Pentax 12-24 and the Sigma 10-20 here. Both are great lenses. The 2mm difference on the low side does make a difference in both width and quality (my opinion). The width is wider, however to gain the width you need to bend that light into the sensor, and that increases edge distortion. So, for me the compromise was the 12-24. I find it both sufficiently wide (at times a bit too wide) with a sharp, crisp image - with little distortion.
The Zeiss wide angle lenses are wonderful, extremely sharp - along with the pot hole left in your budget. As was pointed out in the referenced thread above, Pentax has some additional WA lenses that are out of production that are also extremely sharp - but they take some searching to find and acquire.
All of this bring up a question - Prime or Zoom, since you have a choice between the two in these wide angle lenses. Size and weight plays here across the two lens types. Depends on what you want to carry, and how large a ding you want to put in your checkbook.