1: Moogles aren't classified as a beast tribe.
2: They Predate the Third Astral Era

3: Allags developed Bio-Weapons at Azys Lla so the Moogle being completely useless ruels them out.

This is why I hate the over focus on Allag. There were two Astral and Three Umbral eras before Allag, not everything traces back to them.

How ever, Galecats and Moogles could be loosly related. They're biologically simmiler enough to share a common ancestor.