You don't need a lane bully support. This may just be me, but the only support I have played for about a year now is Taric. If you are good with your support champion and can master their limits you can do well in any matchup. By limits I mean things like exactly how many spells you can cast at each level with your mana, exactly how much damage you can do to any enemy you play against, exactly how many tower shots you can take at each level, etc. This takes hundreds of games on a single champion to learn. I have about 300k mastery on taric to give you an idea of what I'm talking about.
Thresh would be a good one to do this with if you enjoy playing him. A good thresh can help his team in every single game no matter the situation and a great thresh can easily carry from the support position.