Having never disassembled my Mark III to the point of removing the barrel, I can't be 100% certain, but...
I've heard that the actual barrel on the TX200 is very short (something on the order of 10" - 12"). What you see as the barrel is actually a shroud. I would presume the same short barrel is used on both the HC and the Mark III. If so, there should be no accuracy difference between the two.
The HC is lighter, but it's still going to be around 10 lbs. once you scope it. It's a load to carry all day. As far as manueverability, my Mark III is several inches shorter than my RWS 36 that I used to take to the field, so I have no complaints.
I got the walnut on mine, for two reasons: One, as mentioned earlier, AA puts some gorgeous walnut on their guns; two, if I'm spending this much on a gun, another $50 bucks (at the time) seemed like a small investment to get something of which I would be proud to own. I've heard that walnut is lighter than beech, but can't verify that on this gun. Again, given the overall weight, I don't know that you would notice the difference.
In the end, I got the Mark III because I preferred the aesthetics over the HC - obviously a personal preference. Whichever you choose, you can't go wrong!