An assembly includes a monolithic carrier structure and a seal assembly. The monolithic carrier structure includes a carrier base and a support ring. The carrier base extends axially along the centerline. The support ring projects radially inward from the carrier base. The seal assembly is nested radially within the carrier base and axially next to the support ring. The seal assembly includes a seal base, a plurality of spring elements and a plurality of seal shoes arranged about the centerline in an annular array. The seal base circumscribes the annular array of the seal shoes. Each of the spring elements is radially between and connects a respective one of the seal shoes with the seal base.