I am interested in whether the optimal solution involves light-speed (or asymptotically light-speed) expansion, or (for entropy or other considerations) does not actually end up eating the whole light cone.

The form of the expansion has very little to do with the form of the computronium.

Launch von Neumann probes at c-ε. They can be tiny, so the energy cost to accelerate them is negligible compared to the energy you can harvest from a new star system. When one arrives, it builds a few more probes and launches them at further stars, then turns all the local matter into computers. The computers themselves don't need to move quickly, since the probes do all the long-distance colonization.