I wouldn't give up on an accurate barrel.
K98 barrels have a very short (~0.200") shank but you can get away with cutting off one turn of the barrel (1/12" or 0.083") without throwing everything else out of whack.
since you probably arent able to do a chamber cast, have a good look at a fired case and using calipers, determine if threading the barrel in one full turn will clean up the gouges. If so, send it to a gunsmith who can do a chamber cast and confirm.
the fact that the first two gunsmiths you went to did not immediately recommend a chamber cast, as well as the idea that one of them "hoped the headspace doesn't get screwed up too much" is why I suggested bringing it somewhere else. You don't "hope" anything with gunsmithing and especially headspace.
A chamber cast will run $25-50 and the work will be $150-200. If you think thats expensive, wait till you price a Lothar Walther barrel and the work needed to install, chamber, refinish, and affix the sights from your old barrel.
There are no practical rechambering options (30-06 will not clean up a 308 chamber)