Yeah, I considered a lot of those factors.
It would definitely not be used in quite the same way as the traditional FreeSpace support ships. Because the docking would be a little more complicated, pilots in need of support could definitely not simply sit still and have the support ship come to them, line up the docking, etc.. My idea to handle this would basically come down to the pilots heading to the support ship, possibly targeting a subsystem to select which docking port to go to, and then activating some sort of autopilot system. As with normal calls for support, however, it would need to be possible to abort.
Because the support ship is somewhat less mobile than the traditional kind, in general I couldn't see it flying to the pilot first, except if maybe you were the only pilot in range, or the only one in need of support at the moment. It would, however, still have engines, and thus could be a reasonable option in escort missions where the entire operation needs to continue moving forward anyway.
There is one problem I hadn't taken into consideration previously, which is the possibility of fighters being disabled. I have however come up with a possible solution, which is to give the super-support-ship one docking point that would function in the more traditional manner, which would be reserved specifically for that purpose. Another alternative would be attaching one or two drones to the support ship, which would be specifically tasked with engine repair, or simply having a separate support ship capable of system repair but not rearmament.
Obviously, though, this idea isn't as simple as swapping out the Hygeia for something else. It would almost certainly require either some cleverness as far as FREDing, some lua scripting, or the addition of some new features. Really, it's just an 'interesting idea' that I wouldn't mind seeing in FS2, or at least seeing how the effort to get it into FS2 goes. It would make combat logistics more important, and would make the support ship a priority for defense and at the same time capable of fending for itself. It could also make possible the idea of a battle of attrition: even with a larger support ship so as not to have to deal with the [previously bogus] claim that support ships can run out of ammo, losing a support ship could be made into a big deal, and losing a bunch in one battle (if a mission gave you more than one) would easily mean losing the battle as a whole.
Edit: retitled the thread, apparently I had the original project's name wrong...