Starfleet cadets should be sporting Starfleet insignia, not Enterprise insignia, so my vote is for the starburst pin.
Unless, there's going to be a story point about cadets, as part of the familiarization training, being divided up into various dorms, identified as various "ships", like the various houses of a British boarding school (think Griffindor, Slytherin, etc.), and Kirk just happens to be assigned to the Enterprise dorm. THEN you'd have an in-universe, within canon, excuse for using that Enterprise insignia, so long as other cadets are shown with equally distinctive pins representing insignia for other ships, like the Exeter and the Constellation.