First I'd combine two Mandate Archive sourcebooks: The Imago Dei, and Bruxelles-class Battlecruiser for the fleet.
Here's a snippet of The Imago Dei promo blurb:
Not every synthetic mind is bent on mankind's destruction. Some have a higher purpose, a nobler war to wage in the darkness between stars... Aside from their little-known history, you'll find specifications for their bleeding-edge ship hulls. Included are seven sample Imago Dei warships for your own campaign, whether found as derelict plunder, seized by fanatics, or threatening the PCs' homeworld with the guns of a heretical Shepherd Fleet!Certainly an ultra-powerful set of ships, with a fleet of battlecruisers, is a plausible starting point for the mobile base of operations, and it helps combine the idea of 'ultra-tech', 'sentient ship', and 'battlegroup fleet of ships' quite nicely.
The AIs in charge of the Imago Dei fleet would be driven by some noble purpose -- like rebuilding the fallen empire -- which could pose future problems as the empire wasn't all that noble. Furthermore, these AIs also have a tendency to deteriorate as they age, and not in the funny Red Dwarf way either. Early on, though, the AIs can be completely fair and noble with just a hint of what is to come.
Of course, even with a critical mass of people in the fleet hopping from system to system in their quixotic quest, resource, production, and resupply problems are bound to arise. Hence the introduction of the following technologies in top secret areas of the Imago Dei ships: Transhuman Tech. The matter compilers and a status-based economy in a 'post-material' environment are interesting to explore given that there will be a slow, but steady drain of resources on the fleet that needs to be replenished, even with this wonder technology.
Next up: where do the players come in?