Does Galois theory really have no connection with the mission? Consider,
LSE was founded 1895 by four idealistic young Fabians for the "betterment of society". It adopted rerum cognoscere causas, which means to know the causes of things, as its motto.
You could argue, to really understand the structure of factoring polynomials you must study Galois theory. It answers the insolvability of the quintic. Moreover, applications to encryption etc.. must implicitly abound in that Galois extensions seem to be everywhere in number theory (I say this as an admitted outsider!)
For future teachers to know the causes of things they teach, Galois theory has great relevance. Surely factoring polynomials is an essential part of "real world" mathematics. In addition, I would argue that understanding the interplay between extending the number system and solving equations is quite relevant to future math teachers. Galois theory is just another chapter in that saga. The need for understanding Galois theory both in content and historical context is even more evident for the teachers of future math teachers. Is Galois theory a required math education topic? Probably not. But, if you want to be an innovator, you could study the effect of including it in your curriculum. How does it aid their future work? I'm not certain this is relevant to your students, but perhaps you could adapt the idea I sketch here. That sort of education/innovation study seems to be something you could sell at LSE.