Daz, the nearest Federal Medical Center is in Ft. Worth Texas. However, it houses only female inmates. Possibly the closest Federal facility for males in custody would be at Rochester Minnesota, near the famous Mayo Clinic. But if he were transferred to Minnesota, the Federal courts in that state would not have jurisdiction as the crime(s) Osbon is charged with did not occur in/over Minnesota.
The Federal government very often lodges those in custody, after an arrest and before/during trial, in state or local government jails. Robert Reid, the 'shoebomber' was kept at the Plymouth County jail in Massachusetts before his trial. However, once convicted and sentenced, an individual, with few exceptions, is lodged at a Federal prison. All the necessary psychiatric and medical evaluations can readily be done at these state and local facilities.