I would be hesitant about running JB paste through a bore for break in. All you will do is round off the rifling which is not good at all and will reduce the accuracy. Shoot it as much as you like on the day you are shooting then clean it after you are done. Just don't let it lead up a lot if you arent using jacketed bullets.
Steal is inanimate and doesn't feel anything so I am not sure what letting it cool between groups has to do with anything. Its going to be shot hot someday so it should make no difference.
FWIW, I shot my AR the first time, about 60 rounds, then cleaned it later that day. The more I shoot it the more accurate it has become, and no fancy "shoot, clean, shoot, clean, etc" either. I also shot a new Thompson SMG with a new barrel on it, in 90 degree Arizona heat in the desert, over 500 rds with no cleaning or cooling off with zero problem. It was so hot a few times that it would burn flesh just touching the barrel. It looks almost new in the bore still.
Just my experiences.
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Gun Toter