Been wrenching on ARs for a few years, as a hobbyist.  Finally bought my first bolt-action, a Ruger American Ranch 5.56.  Long story short, while trying to install a xeno flash hider, I inadvertently tightened the barrel about 1/8 of a turn into the receiver.  It does still chamber a round but it does take a smidge more effort than I think it should to close the bolt.  

To fix it myself, looks like I'd need an external action-wrench and a set of go/no go gauges.  Littlecrow makes a special wrench for loosening the smooth barrel nut but I have no idea what tool to buy that would allow me to get a torque wrench on that smooth barrel nut.  

Tempted to take it to a local gunsmith but I'm in a small town and not sure if they'd have all the proper tools to do a good job.

Sorry for the non specific answer to your question but I've seen there are aftermarket notched Savage style barrel nuts available for the American. Might be a little more expensive but easier to just remove the old barrel nut and torque down a new one that has notches for whatever the barrel wrench is.

Pipe wrench to get the nut off. Then order a new one that has the flats on it and get the right wrench to tighten.

I had a buddy drop one off with the same issue. Checked and it won’t close on a go gauge. Built a wrench for the barrel nut. Was gonna clamp it in my barrel vise, reset headspace and tighten the nut with the wrench. I’ll weld a socket to it to put my torque wrench on