Deadline...
Does it thump with the faucet open or closed?
If it's just the faucet at the galley, consider swapping the valve stem with the one in the head. If the problem moves, you need a new valve stem.
Otherwise it could be a connection on the cold water line leading to the galley cold water faucet causing the pump to cycle. That would leave water traces at the leak point and water getting into the bilge if so.