An assembly for rotational equipment includes a plurality of seal shoes, a seal base, a plurality of spring elements and a frangible element. The seal shoes are arranged around an axis 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 and the seal base. A first of the spring elements includes a first mount, a second mount and a spring beam. The first mount is connected to a first of the seal shoes. The second mount is connected to the seal base. The spring beam extends longitudinally between and connects the first mount and the second mount. The frangible element is configured to restrict radial outward movement of the first of the seal shoes.