In the interest of keeping away from custom abilities, Guts, Magic Guard and Poison Heal seem the most promising, in that order.
Shed Skin gets an honorable mention, but I think that its luck-based nature ruins it. I want this thing to dependably remove status. Natural Cure is out of the question, as we want our CAP to be able to stay in and get the job done. Synchronize also makes little to no sense. This CAP should be unaffected by most status, and having status stick and then statusing them back would increase the amount of secondary, not reduce it. I would also like to avoid inferior abilities like Insomnia, Immunity, and Water Veil. Levitate is also in this category.
I don't remember who said something about Trace, but are you kidding me? That ability will at best (for our purposes) give us natural cure as we come in on status, and will usually give us an ability that has nothing to do with blocking status/secondary. Arena Trap is kind of useless too. People will know that the CAP is a pursuiter. They will stay in! The only statuser I can think of that wouldn't care about a guts boosted pursuit and would likely switch out is the rare w-o-w heatran. Celebi, Slowbro, Rotom, Cresselia, Roserade etc. will all be forced to stay in.
Guts is number one on my list because it basically allows you to benefit from any status other than paralysis, which ground typing (for the most part) prevents. Boosting our attack when they status us will nearly guarantee our Pursuit has enough power to eliminate the foe, deters people from using status effects in general. This ability is superior to Quick Feet because you don't need increased speed to take on the bulky walls that usually carry status, you need more attack. I am also against Marvel Scale because most statusers are special, and this would go against the offensive nature of the CAP.
Though it is a beautiful ability for the CAP, Magic Guard comes in a notch below Guts because it won't stop burn's attack decrease, which means that pesky rotoms will still be able to w-o-w without fear. That is unacceptable for me. The cool thing is the recoil-less Life Orb, which could allow for mixed sets even if we give him subpar spA.
Poison Heal is a nice option but the problem with toxic orb and this ability is that you will have to bring it in for a turn before you can nullify status effects. This means that you ruin the element of surprise, or come in on a w-o-w before you get the poison up. I think that the element of surprise is too important for this concept, so this puts me off of this ability.
But, we don't need to give this CAP only one ability, do we? Why not give it a choice of two abilities that would, in conjunction, stop whatever secondaries you want?
So, we could give him two of the above abilities, or we could give him one of the above and also give him something like Stick Hold or Klutz to nullify trick. This gives you the ability to customize him to beat the statuses you are more worried about!
@ Pink: Without creating some stupid new ability or giving it Skrappy in preparation for Rapid Spin, there is little we can do at the ability level to reduce entry hazards and Encore. We will probably have to give him a move like Taunt or Rapid Spin to do that. For now we are trying to remove statuses, as there already exist a plethora of ways to reduce status effects at the ability and typing level. Yes, Guts already exists, but there is currently no pokemon that makes good use of Guts + Pursuit to remove statusers. We aren't removing the status per se, but we are deterring it. We will at worst find out that even the perfect status absorber/counter will not cause a decrease in status.