Under the CBA the Rams can play a lot of hardball, three years if they so choose, but it comes with the drawback of them eating his cap hit and the distraction/drama of a great player holding out. I think they might be able to request a cap waiver, someone else can probably shed light on that as I am not certain, but the player and in particular a great player can cause all manner of unpleasantness if he's properly motivated to do so.
I presume AD's representation is claiming he will hold out until he's paid to neutralize the "tag him 3x years" thing. That's strong leverage for the Rams, but not if he is willing to sit out for a couple years if it comes down to that. Given him being willing to not show up period, and the contract impasse remaining in effect, the team almost surely will reach that point where they elect to move him. So IMO the "Rams hold all the cards" argument is a bit overdone, because any player can dig in enough to have the team wash their hands even a future HoF type dude.
And players do it different ways too. You can not show up and be a distraction, or you can be there and be a distraction. Remember our own Eric Dickerson during his final season in horns? He was a malcontent who showed up and would occasionally elect to not play like that Browns game where he went in the game for a play, ran for a TD, and went back to the bench. I don't think AD has it in him to show up and be a distraction. And I also don't think he has it in him to sit out a year or two. So if I had to guess, I would think hardball might be the extra incentive he needs to get his ass in there and sign a compromise. But if not, then he'll be moved relatively quickly thereafter IMO.