Answer

This name of Hashem is unlike the other names.
The other names of Hashem describe Hashem’s powers or actions.
However, this name of Hashem describes Hashem’s existence, His very being.
אהי' אשר אהי' – “I will be that I will be”. Hashem’s existence is permanent and eternal.
Since this name of Hashem describes the very existence of Hashem Himself, not only something that He does, the רמב"ם doesn’t include this name with the other names.