It's worth checking the new battery as well. If it has a faulty, high resistance cell, it will not take as much charging current as it should and the terminal voltage will rise unusually high. The fact that the additional current taken by the landing light holds the voltage down lends credence to this.
If the battery does have this fault then it can quickly be proven by leaving a load such as the landing light on, but without any charge being applied. It will run down much more quickly than it should.