I’d say most of important characters would be probably too busy with interior problems, to do it themselfs. Or atleast I would not be brave enough to ask anyone for their capital ships and main Commanders.
A chance for Urun, Captain of Gondor, to show his quality.
Now, depending on the scenario - how much I know about what is going on.
If I have little information and it’s probably not army of hybrids, I would take Daelaam Mothership. It is primarly self-sustaining huge science vessel, researching the cause would not be a problem, and could provide me with probably enough defence.
If it is combat mission, then I’d love to sit behind Shields of Tal’darim Mothership.
But since Alarak is not a fool, there would be none available. I’d probably insist on Artanis to make 1 or a few Daelaam Motherships more armed and suitable to combat, and then take one of them + a few carriers and other support to help me.
Both cases would be probably mainly heavy mechanised forces - Nerazim Immortals, Dragoons, Aiur Dark Templars, Aiur High Templars (Tal’darim Ascendant would be a gem, though) and a few Aiur Zealots. Supplied by volunteer Purifiers, who are Protoss, not killing machines.
But it mostly depends on the fact that I love Protoss.
Probably best choice for the mission would be Organism Abathur with Kerrigan’s Swarm onboard a Leviathan. Infinite amount of Zergs sort out every finite problem, and with Aba, there would be nothing what could stand on my way.