Jed,
1st, is this on an MGB? If so don't give yourself carpal tunnel trying to remove the float bowl covers, been there done that! Take a few good cell pictures of where the linkage arms connect to the carbs (the linkage has little arms that go into square holes for the throttle and cold start valve. Remove the fuel line hose (hold a small cup under the lines as you remove them). Remove the bowl vents and emissions to the block. Simply take both carbs pairs of 1/2" bolts and washers off the carb flanges. Grasp both carbs and pull them off the studs. With a carb in each hand separate them and let the linkage fall away, nothing will explode or happen with the linkage, it will just hang by the cables . Now take the carbs outside and dump all the fuel out. When you remove the float bowl cover more fuel will come out. Now you can service the carb and stop your leak. It might also be a good time to check the float heights. But if the car is running well, you might want to leave it alone.
My explanation makes it sound fiddly, but I feel you'll be able have the carbs off in about 10 minutes and can do a better job fixing the leak.