BSN has been shooting these rifles longer then I have, so I won;t argue the "cleaning/barrel damage" point except to my belief here: You're pushing a lead-filled copper jacket down the barrel at 2700 to 3100 feet per second, then doing it again within a minute or so....and again...and again. In my opinion, you will NOT damage the bore by cleaning it with a brush, some patches, and some solvent. Unless you're ramming a stainless steel brush (never use them) down the bore that's already two or three calibers oversize, or scraping an ill-fitted end of a stainless steel rod against the bore, then don;t worry. That's improper cleaning and is a whole nother topic.
If you're at the range and shooting 5-shot groups (I reload in sets of five), you'll be loading them one at a time and taking your time. By the 5th round, your barrel will be getting warm, so leave the bolt open and take plenty of time studying your target, jotting down your notes, etc., before beginning your next set. Every so often, I reach over with my left hand and lay it on the barrel. If it's too warm to leave my hand there for a couple of three seconds, I find something to do for awhile. Usually I have another caliber rifle set up on the adjacent bench, so I'll spend some time there.
I clean my rifles after every range session, whether I fired 10 rounds or 40. And I use copper killer each time when starting the cleaning. Usually with my 30-06, I've loaded 20 to 40 rounds, in groups of five different load configurations, so I almost always fire at least 20. If I'm going to fire the remaining twenty, I clean the bore usually, and using all the steps too.
Take your time and the barrel will almost never get too hot. Clean it every time - you won;t hurt it any I don't believe. In fact, you'll do less damage cleaning (make that: no damage) than leaving it dirty for the next session's bullets to scrub on. Clean it well.