I do admire someone who titles their thread "A stupid idea.....". It is so much more refreshing than those who come along and say "I have a theory", when they don't.
Well, Alex, it is, I am afraid, a largely stupid idea.
Point 1: What makes you think we need to get rid of lots of water to counteract global warming? I am not sure why you came up with this. It is true that water vapour is an effective greenhouse gas, but to get rid of the water vapour we wouldn't have to get rid of a lot of water, we should have to get rid of nearly all the water.
Point 2: Balloon implies floating. A balloon of water is not going to float. Perhaps you were just being sloppy and meant a lightweight container.
Point 3: The moon would be unable to lift anything in this way at all. The 'balloon' would still be under the direct influence of the Earth's gravity.
Point 4: If you did want to remove a large volume (and mass) of anything from the Earth's surface, the best way might be with a SKyHook or Beanstalk. This is a 'string' attached to a counterweight, so positioned that the midpoint of the system is in geostationary orbit. Consequently, the 'string' hangs down from space, remaining over the same point on the equator indefinitely.
In practice the string would be composed of carbon nanotubes. Elevators would be mounted on it, running into space and back again. The kinetic energy of those coming down could be used to move those going back up. Power would be required to compensate for efficienty losses, but overall the system would be very energy efficient.
So, you have kind of re-invented the SkyHook, which isn't bad for a stupid idea.