What type of ammunition brands & types have you tried?  If using different ammunition, what were the results or differences?

CY6
Greg Sullivan "Sully"
SLR15 Rifles
TheDefensiveEdge.com
(763) 712-0123

Between the height of the MRO that does not have your check welded on the stock for the same fixed head point shot from shot, and what I suspect is a very high fire rate for such a thin barrel ( which may not have been cleaned correctly to start with, or the upper receiver bearing areas over lubed) can see the problem being more operator error, then rifle error instead.

The Sig 516 specs say this has a hammer forged, 1/7 twist, with a nitride finish.  With this barrel set up, for process of elimination I would suggest trying a different ammunition in a heavier bullet like a 62, or 67, or 75, or 77 grain weights, from a USA made reputable manufacturer like Federal, American Eagle, Black Hills, Hornady, Winchester, Remington, etc.  Sometimes a 1/7 twist barrel will stabilize a 55gr bullet, and sometimes it won't.  It is also not always the weight of the bullet, but the shape of how much of the bullet actually touches the rifling will make a difference in stabilization.  So again my suggestion for process of elimination, try a different ammunition and see what results you may get.

A couple of other things to check for process of elimination:
Manually cycle a few new rounds through the rifle, and when you manually cycle them out of the chamber, inspect the ammunition for any signs of the ammunition getting damaged or deformed in anyway.

Check to make sure the barrel isn't touching anything like the free float tube.

Inspect the forend to make sure it is tight, and hasn't come loose.  Inspect the flash hider / compensator, looking to see if there are any signs of the bullets striking anything.

Since it is a piston action, try turning the gas system down, and see if this changes the group size during shooting, as if the gas system is opened up you may be experiencing barrel harmonics issues.

CY6
Greg Sullivan "Sully"
SLR15 Rifles
TheDefensiveEdge.com
(763) 712-0123