I had the same thing happen to an upper I just bought (not a Colt) and you do need to tighten the barrel nut. While I do not think you need to torque the barrel nut, I did in order to find out what was going on.
Tools needed: multi tool from DPMS $20. DPMS upper block ($25). Molly lube ($3). Strap wrench (had, but IIRC about $10).
First, you will need to remove the gas tube. I just removed the gas tube along with the front sight base/gas block as it all had to come off anyway.
What happened to me was when I tried to install a Phantom FS the barrel moved (about the 1/16th inch you saw), and the M4 feed ramps were out of allignment.
To break the barrel loose, it took 38 ft-lbs of torque. Humm (they had lubed the barrel nut very well).
I tighted the nut back up, at 40 pounds the hole was alligned, but when I tried to install the Phantom again, again the barrel shifted. OK, found the problem.
I tightened the barrel to 75 ft-lbs. The next hole was ALMOST aligned. I back it off and tightened again to 75, now it was closer, but not quite. I did this 4 or 5 times and on the fifth try, at 75 pounds, the hole was aligned, but he feed ramp was off. I took it loose again, put counter torque with a strap wrench on the barrel, tightened until the hole was aligned, screw the ft-lbs this time, and put the thing back together. This time the M4 feed ramps were in alignement with the cuts in the upper and I put back on the Phantom FS without the barrel twisting.
So, the company who made my upper did nothing wrong (which is why I will not tell anyone the name of the Company-they did offer to take it back and check it out but since I had all the tools I needed, crap, I could do that just as well as they could), they were within the torque specs, it was just not tight enough to keep the barrel from shifting while putting on the Phantom and crush washer.
I am in total agreement with HomoErectus about not needing to use a torque wrench for barrel installation, but in this case since it was a brand new upper, I wanted to see what the hell was going on and why my brand new barrel was shifting in the upper.