I would fill the area under the barrel, but if you are more comfortable doing just the sides - go for it. Why don't you look up some pictures of completed bedding jobs on the internet while your release agent is drying. I have never looked for bedding pictures on the internet, but there must be plenty of them. I would bet that 19 out of 20 of them have the barrel supported on the bottom.
Good luck with the bedding. I will check this site in a few hours to see if you need anything, but for now I'm going hunting.