Stoney Littleton at Wellow, near Bath is one of the best and more peaceful longbarrows within driving distance of Weston.
Also, in Bristol, is a site I have yet to visit - a small burial chamber in the Garden of 59, Druid Hill, Druid Stoke, Bristol. An OLD local history book I have seen says that the site "May be inspected [..] at reasonable times" I really don't know what the situation is with that now, or even if it is still there. I really should go and look for myself.
Good luck