I have a T3 in 243 and after about 12 to 15 shots this is what I do.
1, With a nylon brush dip in Butches Bore Shine and scrub though.
2, Patch with a jag until dry.
3, Forrest Foam.
4, With a nylon brush dip in Butches Bore Shine and scrub though.
5, Patch through with a jag until clean.
This works for me and my first shot is as accurate as the last ( thumb hole stuff).
In terms of running a barrel in I was advise by a very good gun smith that cold hammer forge barrels don't need running in. He said that when they fire the 3 shots at the factory to test and certify its accuracy they don't run it in first. Its just not needed. (controversial on this site I know)
bryn