Goose, military politics are a tertiary concern
at the very best. So far as we know, Hayate's only opponent in the brass was Regius, and he certainly isn't a problem any more.
For a more concrete example of what I'm doing, I would like to direct you to a fanfic on my old forum called "Through Troubled Waters". Here's chapters
0-1 and
2.
Note how the author provides no explanation of how or why RF6 is still operating after the JS incident. I've spoken with him and helped him work the bugs out of the first few chapters, and he's mentioned how he started on the fic before the final episodes of StrikerS were subbed, so pretty much everything in the last 4 episodes or so didn't happen or at least happened differently. No Adult!Vivio, for instance, but more importantly, no disbanding after the series ended.
Why? Well why the heck
not? Military politics are concerns to be addressed in fics written by people who actually
care about that stuff. So far as I'm concerned, Hayate had more than enough political clout, as you put it, to keep RF6 together after the JS incident.
Once the 'verse has been spelled out, you're welcome to write a ficlet describing how Hayate talked the bigwigs into keeping the group together, but that's
nowhere on my list of concerns.