Booney,
I do not believe the ABS barrels are any more accurate then any other top end barrel. They shoot very well but so do any other top end barrel that is fitted properly.
Today I shot my 300 AX along side the 338 AX in this post out to 1000 yards and it would be impossible to say one was more accurate then the other.
Heat wise, the ABS barrel just seems to take longer to heat up. That said, hot carbon fiber does not feel as hot as hot steel. Its hard to say what the difference to the barrel liner compared to the all steel barrel is. Some tests have been done that say the ABS keeps the bore cooler. That may well be a fact, I am sure it is but is it worth the DRAMATICALLY higher cost and longer shipping times??????
That is not up to me to decide for customers. IF they want one, go for it, they generally shoot extremely well. I have had a few lately that had problems but every barrel maker from one time or another has had a bad barrel slip out the shop door.
For the money, I would rather take two fluted Lilja SS barrels for the price of one ABS big bore blank.