You aren't the first one to post a discussion about this, and I actually posted my thoughs about it in the other thread but I'll post them here again if you udont mind.
IF Riot decided to make a real group of champions to represent the 7 deadly sins I think that they will all be actuall demons. As of now there are 2 very obvious champions to fit that role which you actually already mentioned. Eve representing Lust and Tahm Kench representing Gluttony. Eve even seems to know Tahm Kench as she has 2 special taunts towards him with one particularly suspicious one: "So how are you river king? I see you haven't missed a meal."
So we have two of the seven already.
If we keep the demon theme there aren't actually all that many. I wont include Shadow Isle Champions, Void Champions or Darkin just so you know. They could fit the Demon Theme more or less but if my presumption considering Eve and Tahm being Demons is correct Eve should have more quotes towards these groups. Also I know Eve used to (maybe) be a Shadow Isle Champion but isn't anymore. If we exclude all those groups there aren't many champions left that could be a demon, so we have to look for champions with ambiguous origins similar to old Eve's lore.
After looking trough all champions the most fitting champions are, in my opinion, Shaco and Jax with Jax representing Pride and Shaco representing Sloth. This may seem weird but let me give an explaination for my assumption.
Jax actually just doesn't really fit any other of the sins and he is a bit full of himself. "Who want a piece of the Champ?" and "Imagine if I had a real weapon!" hint towards it if you ask me. Also Pride is considered the most serious sin in religious terms similar to Jax being one of the strongest champions in the entire league (lore wise).
Shaco is even called the Demon Jester so the Demon part is already there. "But why Sloth though?" I hear you ask. And here is the twist. Sloth (or rather the affiliated greek word 'acedia') doesn't necessarily mean 'lazy' but can also mean 'without care' (according to wikipedia considering the 7 DS) and a crazy Jester seems like the right candidate for that I my eyes. Though 'without care' can of course also mean 'lazy' I think there is the possibility that 'without care can also mean doing stuff (like killing for example) just for the sake of fun and disregard any responsibility. He seems like someone who kills people just for the sake of a good laughter and doesn't give a shit about anything else in the world. It may be a bit presumptious but I just feel like the other sins dont fit Shaco.
Shaco isn't angry enough to fit Wrath, not jealous enough for Envy, not greedy enough for, well, Greed, and not self-loving enough for Pride.
Nocturne could also fit into the Demon trope but I dont really know about him. He would fit Wrath or Envy more or less and he IS called a 'demonic amalgamation' in his bio on the lol wiki but I just dont feel it. There just isn't enough for me to be convinced.
Well, that's my take on the matter.