Levitate really isn't an option for an ability
Qazoo306; we need Sand Rush / Force in order to take advantage of sand, else there's would be no reason to run it with TTar outside of good type synergy.
I do think Grass and Flying are two very helpful types for dealing with Landorus-T. Grass in particular also allows to beat Terrakion and Keldeo, as well as deal with Water- and Grass-types easily, which is awesome. Another benefit of Grass is the immunity to Spore and Rage Powder, helping us better deal with Amoonguss. However, the downside of a Grass typing is that all of Fairies' resistances, Fire, Steel, and Poison, are weak to Ground, meaning it removes the Grass resistance and leaves us weaker to Landorus-T. Steel and Poison are also very difficult to pair with Grass due to the poor coverage they have together, (However, I feel this could be remedied by having a special attacker with Hidden Power Fire, though I'm pretty sure we'd have to wait until a later stage to talk about that) and while Fire has nice coverage with Grass, it doesn't give us a super effective option against Fairy-types, and also leaves us weak to Rock Slide.
If we feel Fairy-types are big issue, then
Grass / Fire or
Grass / Steel would probably be our best bets. Grass, as I've already mentioned, gives us a way to deal with many of the threats laid out in this thread so far. Fire typing gives us a way to threaten Steel-types while still being able to switch into Fairy-types, but has the downside of being weak to Rock Slide and having no super effective STAB for Fairy types. Steel typing gives us a way to offensively threaten Fairies, though it leaves us vulnerable against opposing Steel types.
One issue I do have with these typings, however, is the fact that Landorus-T is not threatened offensively. This means that out CAP would either need A. Enough (Special) Attack to OHKO Landorus with a STAB Grass (or Fire) type move, or B. It would need some kind of coverage move, like Ice or Water, to hit it super effectively.
Another route we could take, brought up by
Pwnemon, is Water. Water allows us to offensively threaten Landorus-T and Terrakion, as well as deal with Steels and opposing Water-types quite easily. However, Water leaves us no way to switch into Ground, Fighting, or Fairy, and means we'll be walled by Grass-types.
One options I like with Water are
Water / Flying. Water / Flying gives us an immunity to Ground and a resistance to Fighting, and would give us a way to offensively threaten Grass-types and Keldeo. However, the downsides of this would be no real way to deal with Fairy-types, as well as a weakness to Rock Slide, though we would have an offensive STAB move to help deal with it. While this typing has been explored in numerous Pokemon, none of them have a really powerful Flying-type STAB move, which we could give our CAP. Hurricane is an iffy STAB option, Air Slash is weak as hell, and going with something like Brave Bird means that we'd be hurt by Intimidate, burns, AND recoil damage. I'm not very familiar with CAP, so I'm not sure if the addition of a new move would be possible. If it would be, I think a better Flying-type STAB move would be a great thing to give this CAP should we go with Water / Flying.
The main problem is that Tyranitar is weak to both Ground AND Fairy, and those two types in unison get perfect type coverage. This means we have to decide which one is more important to cover when creating this CAP, making it difficult to choose the best typing. A nice thing is at the movepool stage, we might be able to add Hidden Power type to help with coverage issues, especially those of the first two types.
Overall, while I think Water / Flying is in theory the best type, the lack a reliable Flying-type STAB move means that one of the first two types would probably be our best bet should we give it the correct type coverage. (However, if the ability to give it, say, a 90 BP special Flying-type move is possible, then I'd definitely prefer Water / Flying for the CAPs typing.)