In my experience, lightweight barrels have performed more consistently with full length bedding, or a pressure point near the end of the forend. There are several variables that will make differences though, as others have stated. A good solid laminate, kevlar, or fiberglass stock is a must if you plan on bedding the full length of the barrel. The cheaper plastic stocks are not stable enough.
Varmint/heavy barrels do perform better when free floated, because they are much stiffer than a pencil/sporter weight barrel. Properly full-length bedding a sporter barrel will decrease vibrations and barrel whip, and give you more consistent accuracy. Someone else mentioned this is what Melvin Forbes does with the NULA rifles, which are incredibly accurate. But the kevlar stock he uses is stiffer than steel, and helps decrease vibration.
Sorry for rambling. Anyway, I would start by adding a slight pressure point near the end of the forend, and see if that makes the rifle shoot more consistently. I've done this with cardboard or plastic before, just to see what the rifle likes before using bedding compound as a permanent fix. Good luck.