I "broke" in many benchrest barrels when I was shooting competition...jeez, it sure is no fun, boring as killing ants on your front steps with a hammer.
On these very high quality benchrest barrels here is what I would see:
*fire one shot, brush and apply Sweet's 7.62 Copper remove
it is very important to notice the amount of blue'
on the patch when you patch out the Sweets
*fire second shot, repeat procedure above, paying particular
attention to the amount of blue on the patch
when you patch out the Sweets
*fire third shot, and clean
fire forth shot and clean, Blue on patches at this point
should be much less than the first shot and second shot
*fire fifth shot, and clean...watch blue
fire three shots and clean, watch blue, save first patch
fire three shots, clean and inspect for blue
fire three shots, clean
fire three shots, clean,
fire three shots, clean, compare blue patch on first 3 shot
fire 5 three shot group and clean, watch blue,save patch
fire 4 more 5 shot groups, clean and watch first blue patch
during the 3 and 5 shot groups, I am usually working on a load.
The most important shots are the first 10 shots and more importantly the first 3 shots...you can really see the amounts of blue decrease on the patches after the first few shots.
I would suggest a rod guide from either Lucas or Neil Jones that has a bushing that fits on the rod to keep the rod centered in the bore, if you do not have that bushing, then you are adding wear to your throat area with the friction of the cleaning rod. Plastic on steel has more friction than steel on steel...check it out in a Machinist's manual.
Your 308 should really surprise you with tuned handloads. It is very difficult to "break" in a factory barrel, however some of the Stainless barrels on Rem, Sako, and Rugers are quite impressive, compared to their blued counter parts.
I have developed two different lapping & polishing solutions for my own barrels, I do not sell them. You can achieve a very good polish in your barrel by polishing your barrel with JB saturated on 10 patches on a very tight punch type of Jag, short stroking the patches back and forth in your barrel, paying particular attention to the area just in front of the throat.
Never, never use Rem Clean in the yellow tube or Flitz in your barrel, they will lap the lands round on the corners.
I might add, that when I quit shooting Benchrest, I also quit breaking in barrels with the fire and clean method.
Good luck and best wishes with your new barrel.