It’s actually interesting to think about how much Oberon and the Shadow Fox know about Inoda and Asuna. For example, there was that event where they used Alice under hypnosis to lure Kirito in and wait for Asuna to show up, in order to frame him. Oberon voiced that idea himself. I think he and the Shadow Fox are working together, but for their own goals, not on behalf of the Council as a whole.
That might explain why they don’t have complete information. If they did, Oberon would likely try to frame or eliminate Inoda to get closer to Asuna, not just focus on Kirito. And how much the Council itself knows about these relationships is also unclear. We know that Sinon is aware of it from what Suguha told her, and Liz knows too. But among them, only Sinon has a direct connection to the Council, and it’s not certain if she’s shared this information. Personally, I don’t think she has.
Agil knew those two were planning something against Kirito, but that doesn’t necessarily mean he knows about Asuna’s involvement. So we end up with this funny situation: they’re trying to take Kirito down because of the myth of his perfect love with Asuna while totally unaware that she’s actually cheating on him with Inoda.