I think my head is going to explode............(updated, sort of, post 32)

[Patrick Chadwick]

The rifle is now wearing a Vltor stock and Vltor M14 rail.

That is, I think, part of the problem. A rifle should not "wear" anything. It is not a tailor's dummy. And the "clothes" affect the barrel dynamics.

Now for Grandpa's fundamentalist point of view:

Get rid of the plastic flimflam, ground-tracking radar, satellite telephone, automated teamaking facilities and whatever else is hanging on that long-suffering barrel and turn it back into the original GI configuration as far as you can.

Then try it out with ammo that is also as close to GI ammo as you can get or reload.
You may be positively surprised.

Some of these fully tricked-out modern rifles remind me of the Starfighters that were bought by the Germans way back heaven knows when. They were offered in various options: high-level interceptor, ground attack bomber, long-distance reconnaissance etc. etc. And the Germans said "Yes" to everything. In one model. The result was a flying emporium of all possible functions that crashed notoriously frequently, sometimes because the pilot pressed the wrong button and ejected himself. On one occasion while the plane was upside down, doing a roll over the runway at an air show. Hammered into the tarmac at several hundred mph, there was not much pilot left to bury.

Seriously, I think your barrel is suffering from bad vibrations. Get back to basics and start from scratch. If you can't find specific info for the M14, I imagine that the tweaking tips for the Garand will illustrate the principles quite adequately.