It's something I've been trying to iron out for quite some time... mainly since I've wanted to make aura-wielding OCs who aren't pigeonholed into using Riolu/Lucario as their ace.

Some thoughts I already have:

• 1. The Cutiefly line could arguably count since the Pokedex says they can detect the aura of living things. And Medicham has apparently sensed auras in the anime according to Bulbapedia. But this doesn't quite translate into their movepool(?) because they can't, for example, use Aura Sphere. Do they count?
• 2. But on the other hand, does the ability to use Aura Sphere necessarily mean that Pokemon can use aura in general? By that logic, Togekiss, Clawitzer, and some Squirtles are Aura sensitive.
  • Other Pokemon who qualify under this definition: Hitmonlee, Hitmonchan, Blaziken, Lopunny, Mienshao, Hisuian Decidueye, Kommo-o, an assortment of legendaries.
• 3. The Ralts line are basically top-tier empaths in the Pokemon setting. But is this psychic powers or aura? And Gardevoir wasn't capable of learning Aura Sphere until Legends.
  • The Hatenna line are also powerful empaths... though they don't have any explicitly aura-based moves. Same conundrum.
• 4. Can Pokemon capable of learning Dark Pulse wield aura? It's apparently an aura-based move.
  • I've also heard theories that Bite becomes Dark-type because it's basically an otherwise ordinary bite infused with malicious aura or something like that.
  • And would Galarian Moltres be the strongest aura manipulator if it can potentially destroy auras?

1. Detection and manipulation are different things. From a 'lore' perspective, to me, it reads like they can do the former but not the latter. So on a team of aura Wielders I would say they're not necessarily qualified.
2. Yes. If they can use Aura Sphere, by necessity it means they can manipulate Auras. You could argue this is not necessarily true as there's no reason Luxray or Tyranitar should be able to wreathe their fangs in both Ice AND Fire for example, so sometimes moves obviously get made available to fit their desires from a power level or coverage standpoint. But it's my opinion we're forced to accept those abilities as reality, and all those you listed and more can manipulate auras and make a sphere of aura-based energy, so clearly they can manipulate auras.
3. I think it's fair to say that Stored Power is evidence of Aura Manipulation, and thus the Ralts line, among others, are Aura manipulators. However Hatenna and so forth reinforce the idea that Psychic powers in general and Empaths (detection) are not the same as Aura Manipulation. So I'd say Ralts is in, Hatenna is out, just mho.
4. Yep Dark Pulse says "the user releases a horrible aura imbued with dark thoughts". So clear evidence of Aura Manipulation. Crunch is just "The user crunches up the target with sharp fangs", and has no evidence of being aura based in its description. My question would be, if it's an otherwise ordinary bite infused with malicious aura, why isn't there a normal type Bite instead? What does a Bite look like which is NOT infused with malicious aura, and why isn't the malicious aura part in the description?

As far as Galarian Moltres, it depends - it clearly has an incredibly potent aura, no doubt. But the entry says "This Pokémon's sinister, flame-like aura will consume the spirit of any creature it hits." If it's ANY creature it hits, does that actually imply control, much less manipulation? Is having an incredibly Potent aura enough, or must you be able to Manipulate it and Wield it? Depends on how you look at it, but certainly I'd say that it'd be tough for anyone to argue against it serving as the Ace on an Aura based team.

Just my musings, obviously at the end of the day it's your concept and you're perfectly justified including or excluding whatever Pokemon you want. I hope the perspective at least helped clarify your thoughts if nothing else though, cool concept!