If you have participated in PowerPlay, you know that it is setup so that certain gov types reduce the amount of Fortification work required and other gov types increase the work required. You can see the specific gov types that are good and bad listed for each Power in the PowerPlay UI on the OVERVIEW screen. As the OP mentioned, only three gov types help the Aisling Duval Power and they are all independent.
This has nothing to do with either lore or role playing, this is a key game mechanic designed into the PowerPlay feature.
So now with the new Squadron feature, it is IMPOSSIBLE to support the Aisling Duval Power and a Faction that helps the Power you have pledge to. This is obviously a mistake made by FDev and they need to fix it.
Fortunately there is a VERY simple fix. FDev needs to remove the link between the supported Faction and Power in the Squadron setup. You should be able to choose to support any Faction type and still pledge to any Power. Having them linked the way they are today, makes absolutely no sense (as far as game mechanics is concerned).