From the original game with it's mix of anything from corporate state to anarchy and the GalCops patrolling the lot I always imagined them as a separate agency that hold allegence to no side but the rule of law so no matter the faction every system paid towards a sort of Interpol of the spaceways and was assigned a number of cops dependent on the contribution they made.
this helps to explain why it's so hard to get away with anything in the richer CS systems as they can afford to contribute the most to their policing while poor anarchy planets struggle to afford much as their method of government doesn't enable them to collect a whole lot of credits in taxes, it also explains why everywhere you go the cops all fly the same ship, otherwise the police force from a poor planet would be using a rag tag collection of patched up relics
with the advent of long range cobra's in ED starting randomly on the frontier of known space to explore the galaxy I would imagine an Explorer Corps being set up in the same way as the GalCops, with all sides sharing the cost and no one power reaping the sole benefit of new discoveries/trade routes and the like.
so, should I be lucky enough to have the option I'd join the cops for a tour or two followed by the explorer corps for a bit - both being outside the factions it should raise my profile with all of them and allow me to pick and choose my options from there