Somehow, I find it very, very
hard that Earth's local space military could ever have accepted the name "UE Space Probe Agency". Militaries do not
accept wussy, sissy, completely nonmilitary and nonpatriotic names like that. They just don't.
It is much easier for me to think that Starfleet would work together with a "peaceful" UE organization dedicated to space exploration, closely enough to warrant a joint seal, and that this peaceful organization would somewhat predate the space combat force because probes like Friendship 1
are probably much easier to execute in practical terms than credible space combat starships, early on in the history of starflight.
Having Kirk work, at least part time, for a peaceful Earth-run space research organization sounds quite acceptable, OTOH. And this is something he'd be happy to declare to a fellow Earthling from the 1960s, as opposed to confessing to his day job as the minion of an interstellar and thus essentially foreign military.