Before bringing in a verisimilitude/'realism' argument, please consider what it is that you're asking for. When you ask for the ability to relocate a starbase, you are not asking for something remotely comparable to shifting the orbit of the ISS slightly. You are talking about moving a massive structure a distance which is at minimum on the order of 1 AU and may well be several lightyears, and while the fact that shipyards can do it does help the case that starbases perhaps could do it, it is by no means guaranteed. We have no real idea of the relative sizes or structural integrity of shipyards and other starbases, nor do we know whether or not there is some limitation in GalCiv FTL technology that prevents sufficiently-large structures from being moved (granted, we did have Terror Stars in GCII, which were about as mobile as shipyards are in GCIII, but those were, according to the listed masses of the stations and terror stars, roughly a quarter the size of a fully-upgraded economic starbase, or about as large as a station with ~5 modules, and the hull form of a Terror Star looks much more suitable to surviving the stresses of moving the structure than any type of starbase in GCII or GCIII does).
When you ask for the ability for a starbase-mounted weapon system to hit targets at the edge of the starbase's area of effect, you are asking for an FTL-capable weapon system, one whose FTL drive is capable of speeds at least comparable to and possibly significantly greater than the speeds which can be attained by standard vessels. After all, in the second-worst case scenario, the station's sensor bubble and the station's area of effect have the same radius (7 tiles), so the weapon needs to be capable of an FTL speed roughly 7 times greater than the FTL speed that the target is capable of (the missile needs to be able to cross 7 tiles in roughly the time it takes the target to cross 1 in order for the station to hit the target at the edge of its area of effect; the worst-case scenario is worse, because a station without sensor upgrades or a spotter is not even able to see targets beyond 4 tiles away, but its area of effect extends out to at least 5 tiles from the station, indicating that the station can attack targets that it doesn't know are present). This drive, which may have to be significantly faster than the drives found on regular ships, also needs to be sufficiently cheap to be viable for use on disposable ordnance, sufficiently small for it to be reasonable for the missile to be essentially immune to antiship weapons, and sufficiently small for it to be reasonable for the station to launch enough that there is a reasonable chance of the weapons penetrating the missile defenses of the targets (i.e. the system needs to present a credible threat to the target). The support for this being possible is rather lacking within the game; Tiny-hulled vessels have utility issues due to how difficult it is to give them reasonable offensive and defensive capabilities while maintaining enough strategic mobility to be effective over any significant distance. When you ask that the station's interceptor complement be permitted to engage the target, this on the surface appears somewhat more reasonable ... but these are the same kinds of fighters that are carried in carrier modules, craft which are presumably not FTL-capable in the first place, or if they are FTL-capable, they're likely too slow to keep up with the fleet in the first place. These fighters would need to be capable of the same kinds of speeds as the missiles in order to intercept interlopers, which would appear to be inconsistent with the rest of the game - I certainly cannot make a tiny-hulled fighter with only the modules that a Drone Interceptor has that can attain an FTL speed upwards of seven times greater than the speed of my standard ships; given that it's a straight copy of a Drone Interceptor (or assault fighter or guardian fighter), it's unlikely to attain more than perhaps half or a third the speed of a mid-game mid-size heavy warship seeing as it lacks any hyperdrive modules, which is at best barely adequate for a station with unupgraded AOE and fully-upgraded sensors (the target crosses ~11 tiles in the time that the interceptor must cross ~5 tiles) under the assumption that the interceptors launch as soon as the target enters the station's sensor range. You might be able to loosen the speed requirement of the interceptors by assuming that the station always has some out on patrols in positions where it's practical for the interceptors to reach any threatened point on the boundary of the station's AOE in time to intercept an interloper, but we're still hitting the issue of fighters which shouldn't be FTL-capable, or at least not very FTL-capable, and which shouldn't have particularly great endurance having reasonable FTL capability, reasonable endurance, or both.
Giving military starbases something to make them more worthwhile is not a bad idea, but using a verisimilitude argument to justify what you've suggested doesn't hold up to even a cursory examination of what capabilities the suggestion entails when we look at the apparent capabilities of the setting. More justifiable would be something like the military starbase providing nearby colonies with N fighters (the station provides the necessary logistical/command/whatever support through abstracted-out whatever, not station-based fighters rush out to threatened colonies whenever an attack occurs), or military starbases provide nearby colonies with a bonus to planetary defense or resistance because of the logistical support the station offers, or a nonmilitary bonus where colonies under the influence of a military starbase see increased tourism income due to all the off-duty military personnel or increased morale due to the clear evidence of the government's interest in protecting the region, or an indirect military bonus of boosting the (military) manufacturing output of nearby colonies (or perhaps even creating a small amount of military manufacturing for nearby shipyards).
(Also, the "who in their right minds builds a base without being able to relocate it" bit? Just about everyone builds bases without being able to relocate them intact. History is far, far more filled with permanent, non-relocatable installations than with installations that can be relocated trivially, and this includes most modern military installations. You are not simply going to relocate that airfield. You might build a new one and relocate the planes and equipment, but that's not really the same thing; most of the structure of the installation is going to remain in place unless you decide that it's a good idea to take the time and effort to take the place apart.)
Additionally, "influence borders = political boundaries" is something that I despise about GCII and GCIII, and despite the developers' claim that it is not the case that influence borders are political borders but you have in-game options to make that the case, the game does a poor job of supporting that claim; the game provides much more support for the claim that influence borders are political borders, or at the very least claimed political borders. I do not care how strong my influence is in a region, that my influence borders surround your homeworld does not make that region into my space. Yes, this is an extreme example, but it is also something which is possible under the influence mechanics, particularly on easier difficulties. Making it even more the case that influence borders are political boundaries is not something that I will support.