In every vanilla sector (non-random) playthrough I had Hegemony and League come to dominate pretty fast with smaller factions purged out of sector or (if they are lucky and colonized anything away from the center) reduced to irrelevance. Imho they should be buffed.
According to the lore Tri-Tachy is about on par with Hegemony in terms of military potential, yet because Tri-Tachy has far fewer planets and nothing coming close to Chicomoztoc or Kazeron in terms of size their fleets are weak and can't stand up to invasions or successfully invade themselves. During the AI wars Tri-Tachyon had inferior numbers, inferior morale, lower quality of officer corps but they made up for it with tech quality, superior intelligence and superior strategizing with the help of AI which focused on harassment of Hegemony convoys and demolishing their logistics via small fleets of high tech ships and contracted privateers.
This can be ameliorated if a faction fighting Tri-Tachy receives a pirate-like debuff (-X access, -Y stability) for the duration of the war. Maybe only for planets limited to some vicinity of TT systems, maybe for all, not sure which is more lore-friendly. Small fleets of high tech ships would roam systems and attack traders/miners akin to what pirates do. Hopefully shortages caused by low accessibility with strong enough debuff will be enough to even out the playing field.
Luddic Church has Knights of Ludd who swore to protect the faithful, while Church fleets suck ass in general they could even out the field by supporting rebellions without declaring war by e.g. intercepting relief fleets, sending aid to rebels etc and gaining new planets through exploiting the moral high ground. That's how Luddic enhancement mod describes the history of the Cerberus planet and I think it fits with the lore decently.
Diktat... There isnt much lore for me to make up anything really. Problem with Diktat is that they only have one system (albeit strong one), which means that every single raid/invasion targets the same place. Usually Diktat can resist the initial invasion attempts but they get disrupted which tanks their fleet power and the following invasions succeed. Maybe give them more Lion Guard task forces to help out with invasions and defences?