I have an LSS from a few years ago that has a pretty light barrel on it. It had the front pad on the stock that put upward pressure on the barrel. I had the action pillar bedded, and free floated the barrel. It shoots about the same as far as overall accuracy. However, since the pad is gone, when I take the gun apart and then reassemble it, I don't have to rezero. With a barrel pad I found that small, but significant changes in the zero would occur, probably due to variable torque on the action screws. The other big change was the accuracy with a bipod. When I had the barrel pad on the stock, accuracy from a bipod was not good. It wasn't until I free floated the barrel that I was able to get good groups with a bipod, using essentially the same technique.
My advise, free float the barrel and bed the action. You can always add back in the pad with some bedding compound later on if needed.