A heavier barrel will take longer to heat up. There is no direct correlation between this and accuracy; that is, after a certain number of rounds, any barrel will be hot.
Heat can cause all sorts of problems, but in my experience, they are repeatable. Bill Shehane once had a very experimental rifle that suffered from heat problems. In pretty short order, he learned to hold off a certain amount after x-number of shots (the hold-off grew with the round count). He shot a few winning groups with the rifle. The point is, the heat-related stringing was so predictable he could hold for it.
With the 20-40 rounds you are shooting, any barrel will get hot. I would guess that the barrel channel in the rifle has as much to do with stringing shots as heat per se. One thing that can cause stringing is the difference in temp between the top & bottom of the barrel, and a tight barrel channel can cause this.
I wouldn't run right out & ruin that pretty fit of the barrel in the stock. See if you can find a pattern in the stringing. It will almost certainly be vertical. It the stock is a factor, the first round will print "normal", then the shots will go lower & lower as the barrel gets pretty hot, then move back up after a while as the heat evens. If the groups just start printing lower & Lower & stay there, it could be that something else is an issue.
If the stringing isn't vertical, there is some stress in the barrel doing you in, or some other problem that heat is bringing out, but not directly *causing.*
