Buttons play looks like AJ,KQs(except KQc),QQ+,KQo imo. You are 50% vs this range. If you rank them for likelyhood I would say KQo,AJo are a little less likely because of the preflop action. So you arent looking good vs this range(if you remove KQo you have 25% equity). Now what hands will villain raise? Imo he wont raise any of them. I believe our equity is around 35% vs buttons range. Now how often will we beat BBs hand and how often will he call? Imo we need BB to call every time with a worse hand for it to be +EV (0.35*.1=>1/3). So this isnt a valuebet. Check call.