With all that said, here are some typings that i believe would be good to do the above:

[B]Steel / Flying[/B]

Checks Fairy-types, checks Ground-types, checks choiced Talonflame, and is not weak to any of Aegislash's moves while being immune to Toxic. Obviously, without a lot of special bulk Shadow Ball is going to hurt, but this can be fixed either with abilities, good special bulk, or reliable recovery and decent special bulk. The lack of a STAB to threaten Aegislash is not such a concern, because if we go the set up route, we can just give to the CAP a coverage move that is super effective against Aegislash and can OHKO or dent it after a boost. A CAP with this typing also has good defensivey synergy with Latias, resisting Dragon and Fairy moves, and Latias resisting Fire and Electric moves. The biggest downsides to this typing are that it leaves the core weak to Zapdos, non-choiced Talonflame, and leaves some Fire-type threats uncovered as well (such as Mega Charizard X, which can set up on Lucario or the CAP and OHKO all the three Pokemon after), but the advantages are more than enough to overcome those flaws.

[B]Fire / Electric[/B]

Checks Fairy-types, checks Zapdos, checks Talonflame, and can threaten immediately Aegislash. All those are great traits, and this typing has two additional advantages over Steel / Flying: 1. It is weak to Stealth Rock, making Latias a great natural partner for it, meaning that offensive teams will actuallly want to pair the CAP with Latias, and not just use CAP + Lucario with another Pokemon and 2. Fire / Electric provides great offensive STABs, unlike Steel / Flying, which can't immediately threaten some important Pokemon that trouble our core, such as Zapdos and Aegislash. However, this typing leaves us wide open to Ground-types, and a bit open to Azumarill, which although can be played around if choiced (Latis resists Water moves, and the CAP resists Fairy), and revenge killed by Latias if it is the Belly Drum set, can still be quite problematic, threatening with a OHKO all three of our Pokemon with just its STABs. However, all three of our Pokemon can damage it significantly (Latias with Thunderbolt, Lucario with Iron Tail, and the CAP with Electric STAB), meaning that it can at very least be played around, even if it means sacrificing one Pokemon of the core to take it down.

Water / Ground sounds decent, but without a Fairy resistance you just can't switch into Pokemon such as Mega Gardevoir, Choice Specs Sylveon, Choice Band Azumarill, and Mega Mawile, all very big threats to our core.