Dear Jesse,
you are right! I first hesitated the detour going from mesh to graphics and back to mesh, but maybe this is the only way:
sphereMesh = DiscretizeGraphics[Sphere[]]; (* ... as an example *)
spherePrim = MeshPrimitives[sphereMesh, 2];
sphereBoundary = BoundaryDiscretizeGraphics[spherePrim];
Through[{Head, Volume, RegionDimension, RegionEmbeddingDimension}[sphereBoundary]]
(* Out: {BoundaryMeshRegion,4.152965792075726`,3,3} *)
Thank you four your suggestion! Regards -- Henrik