If I change my civilizations capital, won't it lower the loyalty of my plants around the original capital?
it will, but you might find a place where the overall loyalty would be higher. if for example your homeworld is at the edge of the map, then relocating it to the center would profit the far off colonies and would not overly impair the ones that were closer, because they are still fairly close. it depends on the case, but more often the relocation won't be utterly dramatic, maybe 2 - 3 jumps from your original hw.
so again, it could lower the allegiance of the worlds near your original hw, thus pick your new one carefully.