They not legitimizes each other at all. How can CAU legitimize them killing civilians with FD? "Sorry, I know you're innocent, but let me execute you because the existence of some other group you have nothing to do with" - that's rich. Also, while Protectorate is posing as protectors of civilisation, FD are at least has an honest approach. The attack on the lab fits into their agenda, while the CAU is against everything what the Protectorate tries to resemble towards the public.
And I have to come back to the reasoning of my previous post, Protectorate had all the means of defeating FD with legitim ways without civilian causalties and nobody would've question their PR machine. Yet they choose to try out mutagen on the public. The fact that both would result in victory yet they went for the option that involves unnecessary massacre, clearly says how much they care about the people.
FD and Protectorate could only legitimize each other if the latter was only rivals, but essentially the same kind of group as FD. Protectorate however decieves the public, pose as a helping hand then betrays them and kill them. In a way, FD was more useful for them then a setback as antagonizing FD certainly helps pushing their false propaganda through. They could blame them as the root of all evil, even the ones themselves are causing. Zero responsibility.