I have had a misaligned neck pocket on a couple of builds. Both times on ES style bodies. Not off by much but enough so that the hole for the bridge was not 100% on the center-block. Most ES's with book-matched veneer have a built in deceptive centerline... You are so right that it does not have to be off by much to cause a problem.

I use something like your fishing line approach (I use nylon string most of the time). I usually do it with the two outer strings, and then check it with the two inner strings. I try to get the two outer strings to be equally close to the edge of the fingerboard all the way down the neck. In dry-fit, I try to get them to line up over the outer pickup poles as well. When I think I have it pretty close, I sometimes try it again with the two center strings. In this case I do exactly as you suggested. I look to see that the strings are equally close to the dots on the fingerboard. I think it works because you can see when things are off even by a little when you are looking at it over a 25" (or 34" on a bass) range. Not sure how others do it. My method is not particularly scientific and it's certainly possible that there's a better way.

It's also very easy to get the bridge slightly off perpendicular to the strings. In this case, I try to err on the side of making sure the treble side is EXACTLY in the right place, and if the bass side is slightly further oblique to perpendicular, that can only help with intonation, right? ;-)

If you need something thin to shim with, I tend to use veneer. For a few bucks, I got what has amounted to a lifetime supply on ebay. I have even used strips of it in varying lengths to make very thin wedges.