Should I mention my university name when publishing a paper if I'm tenured teacher but not researcher?
The fact that this research isn't part of your job shouldn't prevent you from giving your employer as your affiliation. A paper by David Pincus about certain weak forms of the axiom of choice in set theory (J. Symbolic Logic 67 (1997) 438-456) lists his affiliations as "Department of Anesthesia, Harvard Medical School, Massachusetts General Hospital, Cambridge City Hospital". I can assure you that none of these anesthesiology departments paid David to prove theorems in set theory; he did this work in his free time.