Class C
A 'Class C' stellar engine combines the two other classes, employing both the propulsive aspects of the Shkadov thruster, and the energy generating aspects of a Class B engine.
A Dyson shell with an inner surface partly covered by a mirror would be one incarnation of such a system (although it still suffers from the stabilization problems as a non-propulsive shell does), as would be a Dyson swarm with a large statite mirror (see image above). A Dyson bubble variant is already a Shkadov thruster (provided that the arrangement of statite components is asymmetrical); adding energy extraction capability to the components seems an almost trivial extension.
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