My inclination would be to leave it as it is, but get the documentation on it, that it was brought back by the Sgt and then used as his hunting rifle. That is where the real "human interest" will be with the rifle.
You certainly could "bring it back" to the original military condition, as it doesn't look like it was drilled and tapped or otherwise brutalized, and it has all original metal parts by the look of it, except the barrel bands and sight hood. However, in the end you are going to have a bcd43 Mauser with matching metal parts, but a non-matching, non-original stock with non-matching barrel bands, and it will only be worth a bit more than what any other mix-master will be. It will never be a super valuable collector piece. If you spend the money to track down a full stock and barrel bands for it, then put it on a rack next to a bunch of the "RussianCapture" stuff, aside from the fact it's matching parts, it wouldn't be readily discernible.
There are "bubba" sporters and there are well sporterized rifles. This is one of the latter. It is not a typical hacksaw nightmare. I'm not a big fan of carved stocks myself, but that is a very well done sporter job, clearly done for someone who liked the rifle and wanted to keep it. If I wanted a nice 8mm Mauser deer rifle that would shoot well, stand up well, and look decent into the bargain, that would be one I could see going for, and it will probably always be worth more as it is now than it would be resurrected to mil-spec. However, that it just my opinion and others may disagree.
I just think it's neat that the guy brought it back, and you know who it was, and that he obviously cherished it and used it.
Ed