I don't remember the model number, but Beech flew a prototype of a "Jet Mentor" in this time period
520-C-1. This projected turbojet version of the T38 has a design thrust of 1,870 lb with an sfc of 0.927 lb/hr/lb. Production awaits a Navy trainer or target application.
Is assymetric thrust much of an issue on small twin-engined jet trainers?According to the "Der Flieger" magazine 1953, it was the Ryan (Model) 59 :