Barycentric/Orbital stability of a Dyson Sphere and Central Star

Can a Dyson Sphere and a star inside of the Dyson Sphere, both have a "stable" orbit around the barycentre of a system where they are not the most massive objects? And/Or Can they both orbit together around a larger body? These are effectively the same question but with the Barycentre Inside Sphere and the Barycentre Outside Sphere respectively.

Since I'm not expecting any of these cases to be completely stable, so where/if possible I would like to know details about the differences in how much work it would be to achieve 'active stability' for such arrangements.

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Barycentric/Orbital stability of a Dyson Sphere and Central Star

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