Hello Arizona
I think that you have to define what level of single shot you are looking at. In some cases the single shot is as accurate as the barrel, but these days I don't think that anyone is consistantly making bad barrels.
In most cases it is the design itself that keeps it from becoming accurate.
Besides the firing mechanism (striker or hammer), the way in which the foreend is attached and bedded to the barrel can make a big difference.
I worked on a fellow's Ruger #1 in 30-06 that wouldn't group into 2", until I added a shim inbetween the foreend hanger and the stock. Groups droped to 1.25 immediately.
I have one in .375 H&H that consistantly shoots into 5/8" with Federal and Winchester factory, but that gun is a few pounds heavier than the 06 and I have to shoot it with a sling to get that accuracy.
The Schutzen crew shoot single shots like the Miller, but those guns aren't light either.

The cartridge that you put in it makes a difference (lighter chamberings seem to be more forgiving than heavier ones from an inertia issue.)
A good trigger can make a great deal of difference especially in a light rifle.

My choice would probably be a #1 RSI in 7x57. I would fix the trigger then shoot it to see if any changes have to be made elsewhere. The mannlicher stock would be there to protect that skinny barrel from dings

I'm sure that I missed something here but I think that's most of it.

Cheers.

Tom