To me, the term 'walking varminter' implies the ability to take small-to-medium game and varmints out to perhaps 350 yds while at the same time remaining light enough in weight to be comfortable to carry for several miles.
If you want to shoot deer or small game at close range then iron sights are fine, but if you want to shoot varmints then the accuracy and sighting requirement levels rise dramatically. A scope of 6x is perfectly adequate for shooting out to 350 yds on objects even as small as crows, while iron sights won't do much good on a crow at much beyond 90-100 yds.
Scope mounting is a problem with any period rifle having any monetary value or charisma, and the Kurz has an abundance of both. I agree with the previous poster who suggested a Unertl or Lyman external-adjustment scope with the bases mounted on the barrel, unless the receiver has already had some sort of mount installed or provisions for one such as a square bridge. A side-mounted offset scope a la Krag or levergun would also possibly eliminate the need to alter the bolt.
I personally would choose (for a Kurz Mauser walking varminter) a relatively short-&-light barrel, say about 24" with a muzzle dia of 0.600" or less, and in some sort of period chambering such as the 6x57 Mauser or 22 Newton or similar.
Sling swivels are a MUST on any rifle intended to be carried for any distance at all (except for a DGR of course), especially in hilly country. Also a leather-covered or rubber butt pad is desirable to prevent slippage.
I personally like DSTs for use in the field. Yes they're slightly less accurate from the bench but it's been my experience that they perform better on (non-DGR) game than the typical single-stage trigger, due to the shooter's much-improved letoff timing.
Regards, Joe