No no no, do not add a PVP element to the Pilgrims of Dark. There need to be at least a few PVE only covenants, and since the covenant of champions requires killing invaders to rank up that pretty much leaves the Pilgrims of Dark as the only one. The only thing I think that needs to be fixed about it is that the guy should not talk so much each time you enter the areas (costing a human effigy to enter is fine imo) and it would be nice if you didn't have to fight through a dungeon every time you wanted to face the darklurker.
As for the others definitely lower ranking requirements for Blue Sentinels and Brotherhood of blood, I would say get rid of the ability to lose progress for the brotherhood as well. I would keep duels as a viable method of ranking for each though, as they're fun and removing them as useful for ranking up would really reduce how many people spend time doing them. Way of Blue is mostly ignored because invasions are so rarely a threat, so maybe give the brotherhood a full red eye orb as well. Also red sign soapstone invaders should be made to count up towards rankings for both the brotherhood and the blue sentinels, they really aren't very useful otherwise.
Bell Keepers is fine by me, maybe buff the armor a bit but other than that its about perfect. Its got some fun completely optional PVP areas and gives a nice reward for each victory, making blue sentinels able to appear against them wouldn't do anything as if people want to camp against bell keepers they already summon white phantoms all the time.
Rat King covenant is too unbalanced in favor of the Rat, especially at doors of Pharos, it would be more interesting if the rat had to actually direct the enemies and traps in the area to make a deadly maze the trespasser had to navigate but as it is its not too bad as the trespasser doesn't lose anything but time.
Covenant of champions is mostly fine but I do think there should be more ways to get awestones (maybe by beating certain bosses as well?) and lower rank up requirements.
Dragon covenant is also fine, though as you said it would be nice if the dragon scales had more uses. Heirs of sunlight are similarly fine, though both suffer from a problem that almost all the covenants have that is the rewards kinda suck. Heirs of sunlight has exactly one good reward with sunlight spear but its pretty restrictive in terms of stat requirements. The dragon gear is better but the sword you get is pretty worthless. Other covenants suffer similar problems and even worse the rewards very frequently aren't really tied to the covenant in any real way, there are no spells you need to be in a covenant to cast, and some of the equipment doesn't even have much in the way of lore ties to the covenants, it just makes them all feel rather weak.
As a final complaint I've run into way too many gank squads as a blue sentinel, I'm not exactly sure how one would fix this but even so its really annoying especially since its already so hard to rank up in the covenant. As other have also pointed out there are a few weapons and maybe some spells (dark fog is particularly annoying) that could stand to be nerfed for PVP.
Edit: Also for everyone whining about hosts healing when you invade them, deal with it its as much part of the game as invading is and its your job to stay on them and not give them enough room to heal up. Is it a pain when they do? Sure, but they're just making use of every tactic at their disposal the same as any phantom that lures a player into enemies. Expecting someone to follow some set of rules you have in your head when you barge into their game (and this is not to hate on invaders, I've done plenty of invading myself) is ludicrous.