Alright, I put it together.
Laffitte and Catarina show up on Egghead yesterday before the siege begins. They know from York that the Marines are coming, but, can't turn off the lasers themselves, and have to wait. SO, they call Teach, and tell him that they can't pick up York and the Seraphim because of the Laser Grid on the Frontier Dome.
Teach goes, "Aight no problem, I'll send Van Augur over to teleport you guys up there. Hide the ship, and wait for Augur."
Van Augur teleports across the ocean, arrives at Egghead, and meets up with Laffitte and Devon. By this point, Saturn has arrived.
Laffitte tells Devon and Augur that they should split up, and take York, and the Seraphim, while also having Catarina take Saturn's image so she can command Seraphim over York.
Augur teleports Laffitte past the laser grid, then waits till Saturn is isolated. Once Saturn is away from Kizaru, Augur teleports Catarina to Saturn, has Catarina touch Saturn, then teleports them away.
Laffitte hypnotizes the Seraphim and York, turns off the Frontier Dome's Laser Grind, and then escapes with the Seraphim and York, having them fly in their boots and shoot down any Marines that bother them.
Blackbeard Pirates escape with Seraphim (except for S-Bear who goes to Bonney somehow), York, and Catatarina having the highest Seraphim Authority by transforming into Saturn.
Edit: Keep in mind Bonney has the highest authority over all Kuma-like Pacifista, meaning all the Mark III's and S-Bear in this scenario, but the Gorosei still have the highest Command Authority over the remaining six Seraphim.