The wife and I just got back from visiting our friends in the tropics. We have gone down there a few times now and were excited upon arriving this trip to see a new BGE on the back porch of the condo. Not a bad way to break it in! We did the lobster two ways.. butterfly and steakhouse style both under direct heat. I don't think you can go wrong either way.
I will say most people will cook the lobster meat side up when grilling. A good way to try is to cook with the meat side down for the first 1-2 minutes then flip and finish with the meat side up. This does not dry lobster out but instead helps sear and cook faster. I cover the tail lightly in olive oil before placing on the grill, once I flip to meat side up I add a cube of butter on top to finish.