BE55 pilot reported cylinder head and oil temperature gauges became inoperative followed by engine power fluctuations and uncontrolled yawing during cruise. Pilot returned to departure airport.

Narrative

Aircraft: Beechcraft Baron BE-55. Pilot experience and background. I am a commercial; instrument-rated multi-engine pilot working toward my Multi-Engine Instructor (MEI) certification.Incident Overview. I was conducting a training flight in Aircraft X with my MEI instructor departing Runway XX at ZZZ. Before the flight; we conducted a takeoff briefing; which covered how we would handle any aircraft malfunctions during takeoff. However; we did not establish who would act as pilot in command (PIC) in the event of an emergency in flight. This lack of pre-briefing contributed significantly to the dispute that later unfolded. Shortly after takeoff; the left engine's cylinder head temperature (CHT) gauge became inoperative; an issue that had previously occurred. During the climb; the oil temperature gauge on the left engine also failed. Despite my concerns; we continued the flight. At 5;500 feet in cruise; while transitioning to maneuvers; the aircraft began violently yawing left and right. The engine sounds changed; indicating a potential power fluctuation or failure. While I was at the controls; my instructor abruptly took over; stating; 'My airplane.'At this point; we had an in-flight dispute over PIC authority; cockpit decision-making; and emergency procedures. Breakdown in Cockpit Resource Management (CRM). No Prior Agreement on PIC During an Emergency. At no point prior to this flight had we discussed who would be PIC in the event of an emergency. While I was acting as the flying pilot at the time; my instructor assumed control without verbal coordination.This lack of clear PIC authority led to immediate confusion and poor decision-making during a critical moment. Instructor's reluctance to [advise ATC]. I immediately recommended [advising ATC] due to the loss of key engine instrumentation and unstable flight characteristics. My instructor refused to [advise ATC]; despite the fact that we had no clear diagnosis of the issue. Given my experience I felt that waiting could put us in further danger so I [advised ATC] myself.Instructor's Loss of Situational Awareness. After taking control; my instructor began descending prematurely as if preparing for an off-airport landing. I strongly advised against an immediate descent; arguing that we needed to preserve altitude for options and head directly to ZZZ.My instructor became visibly frustrated; disoriented; and argumentative about our course of action.Dispute Over PIC and Decision-Making. I asserted my concerns about safety and decision-making; referencing my experience and qualifications. My instructor dismissed my concerns and continued making erratic decisions that conflicted with my approach plan.As the situation escalated; he raised his arm and elbow aggressively; making me feel physically threatened. Incorrect Runway Selection and Unsafe Landing. I clearly communicated Runway XX as the intended landing runway per our initial ATC emergency call. However; my instructor; still confused and disregarding my input; lined up for Runway XY; landing with a 14-knot tailwind. This further compromised safety; given the aircraft's instability and unknown mechanical condition. Why I [advised ATC]. I firmly believed that the aircraft's flight characteristics were unstable and potentially deteriorating. The lack of critical engine instrumentation prevented accurate powerplant monitoring. My instructor was not responding appropriately to the developing situation. My experience and certifications placed greater responsibility on me to ensure safety. As a commercial; instrument-rated multi-engine pilot with time in both the PA-44 Seminole and the Baron BE-55; I recognized the seriousness of the situation and made the only responsible decision; to alert ATC and prepare for an emergency landing.Post-Landing and Final Thoughts. Emergency personnel (EMS; fire; and law enforcement) were standing by on the ground. My instructor and I had a final dispute; and I informed him that this wouldbe our last flight together. I subsequently requested a refund from the FBO; as the lack of aircraft reliability; CRM failures; and overall safety concerns made continued training unacceptable. Lessons Learned. This experience reinforced a fundamental truth in aviation: Safety is paramount. I deeply regret allowing external pressures and instructor influence to override my initial concerns before the flight. However; I stand by my decision to [advise ATC] and prioritize safety above all else.To the ATC controllers and first responders at ZZZ; I extend my sincere gratitude for their quick response and readiness to assist.This situation served as a critical lesson in the importance of pre-briefing PIC roles in training flights; standing firm on [requesting priority handling and] ensuring aircraft airworthiness before flight This experience has made me even more committed to maintaining the highest safety standards in aviation and trusting my judgment in critical situations.