Star Trek: Gibraltar subscribes to the "multiple cores" theory of the Federation as put forth by Star Trek: Dimension. The main portion of this page which concerns us is as follows:
"...on the one hand, there is a core region of the Federation that is only few hundred light years or even less than [one] hundred light years large, where all core planets (Earth, Alpha Centauri, Vulcan, Andor, Tellar) are situated. This area is restricted by the five adjoining powers and has not significantly changed its size for [a] hundred or more years. In the relatively small core region, the whole action of Star Trek: Deep Space Nine takes place, what explains the short distances and travelling times in this series as well as the huge ship fleets that operate in a pretty small volume.
"On the other hand, the Federation also extends outside the space taken up by the "Core Federation" and the five other empires far into deep space - within a region with just the considered diameter of 8000 ly. Here, there are several single planets settled by the Federation, but the area is primarily dominated by agglomerations or concentrations of Federation planets that are far more extensive in size and population and which form autonomous "fragments" of the Federation each. Examples for such Federation fragments, which are detached from the core region and completely self-sufficient with [their] own administration and defense, are the regions around Deneb, Antares and Rigel - just those stars which [are] mentioned as "boundary stars" of the Federation territory with regard to the theory of a large Federation. This wide-ranging and all in all more sparsely populated Federation region was explored by the Enterprise-D within seven years. On the one hand, the Enterprise could travel through the [vast] unknown space between the different core regions without encountering large fleets of the Federation, and on the other hand, she could fly to the single "fragments" - mainly the core region, but also the Denebian region (in the first year) and the Rigel region (near Mintaka, in year five)."
The action of Star Trek: Gibraltar takes place within and around the Antares region, which, according to this theory, is its own autonomous "core." It is self-sufficent, has its own administration and trade. Its arm of Starfleet answers to Command back on Earth, but is largely staffed on its own as a separate jurisdiction to with the authority to make decisions on its own if need be.
The true benefits of Federation membership for these outlying areas are dedication to a common goal, access to technology developed in the core region (they also send new technologies back "home," as seen with the new starship prototypes designed at the Beta Antares Shipyards), and a template for setting up government and Starfleet installations. Personnel, colonists, and merchants go back and forth between both areas--your character could just as easily be from a nearby colony as from Earth. Starfleet ships from the core area may come to visit or help, and some supplies may come back and forth, but for the most part these are fringe benefits.
Make no mistake: this is still the Federation, and it is still Starfleet. The people of this area are just as highly dedicated to the principles of the Federation, if not moreso, as the people back on Earth.