Playing against it wouldn't be that hard. Most valk ships seem to have problems controlling their flux when firing and the Vatican is so slow to turn you can get a cruiser like the aurora and just sit right in front of it where it doesn't have any heavy guns and pound it down. The vatican only really becomes absurdly OP when a player is using it, although I imagine it is quite strong in ai vs ai battles too, since the ai wants to flank things and that is a bad idea with the vatican. You might actually want to leave the vatican in a few fleets though, multiple mods have super capital ships like it running around and without it the valks will get walked over by several factions.
The valks actually feel pretty balanced, they just have a huge amount of front loaded damage with their incendiary ammo. If you take them slow and careful they are pretty easy to pick apart in the AI's hands though. Honestly, the most annoying valk ship to play against is that horrible little jammer ship. That thing drives me crazy, but I love it being in there. It really fills a roll I don't see anywhere else. Valks are one of my favorite factions, and like I mentioned above, don't start feeling broken unless a player has certain of their ships set up in certain ways. Also, I don't know if the RNG is just trolling me, but there are a couple of cruisers/destroyers I never see that might be nice to have.
Yes, the sparrow is obviously made to be absurd. It can walk all over capital ships once you start shifting some of its equipment around. Most of the other council ships though only feel a level or two higher than average. The catch is that they are stronger across the board, they don't have any easy weakness to exploit which makes them an actual challenge for the player, for which I am glad.
The gedune would be massively strong if they actually had a cap ship, but without one they tend to get smashed by sheer size. As long as the fight sticks to cruiser size and down though they seem to win, simply because their ships are so mobile and have such a heavy punch right after moving.
I feel sorry for the antideluvians and the asp, they just get crushed. Anti's have that great armor and health, but without shields it is just too easy to cripple and destroy them. Asps, I think their weakness is obvious.
Nomads/Junk Pirates/Lotus/Interstellar Federation all have something interesting about them, but they are balanced to vanilla so they get walked on.

Shadowyards are really cool, and have some solid weapons, but so much of their stuff just feels flimsy or weak, I really don't play them much but they never put up much of a fight when I go after them.
Blackrock looks super cool, and their 2 cruisers, the stormcrow and nevermore, are monsters, but all the rest of their stuff feels weak. I would rather fight several of their cap ships than a single one of their cruisers.
Nihilic have incredible weapons, and their ships are strong stat-wise but due to their layout they are actually pretty easy to exploit. Nihilic do better on non-Nihilic ships. They just don't seem to have the kind of damage output they need to make them dangerous on their own.
Thule feel like a series of gimicks. Very cool gimicks that can catch you off guard if you don't know about them, but most thule ships seem to just fall apart in a standup slugfest, which is pretty easy to force. Their ships and weapons are creative and interesting but simply can't go toe to toe with most other ships it feels like.
Relics have some very unusual guns and strong shields, but with almost no armor and low hull they tend to get peeled apart fairly easily. The strange guns allow them to pull off some silly wins though. That Zapp

And finally the neutrino. The guys with those horrible little tiny shields. And those wacky infinite range photon things . And those splinter missiles. And those blasters, the really fast ones that make a huge amount of flux but hit like torpedoes. They are one of the few factions that I avoid fighting no matter what I am flying. They just have such massive damage output and they have those shields.... I think everyone knows what I am talking about with them.
If I was ranking factions by overall strength, with default load outs, and no shenanigans, in this mod, it would go something like this. Although keep in mind, some factions do better against one faction than another.
Neutrino, Council, Nihilic, Valkyrians, Gedune, Thule, Blackrock, Relic, Tri-Tachyon, Nomads, Intersteller Federation, Junk Pirates, Lotus, Independents, Shadowyards, Hegemony, Antediluvians, Pirates, ASP
Wow, that turned into quite a little essay.
Edit: I forgot the pirates!
Edit2: Was that council ship the sparrow or the dragonfly, I suddenly am not sure on the name.
Edit3: Upon further consideration, I am not sure how the thule actually win anything. Their cruisers/capital ships and half their destroyers are unable to actually do any real combat. An yet somehow, the ai losses to them. Go figure.