BE36 pilot reported an engine cylinder head temperature exceedance; which resulted in an immediate diversion to a suitable airport.

Narrative

I was on a instrument proficiency flight with a filed IFR flight plan from ZZZ to ZZZ1; where I intended to fly several instrument approach procedures before executing a full stop landing.I was in IMC at 6000 ft MSL when ZZZ Center asked that I climb to 7000 ft to accommodate non-radar traffic. I entered VMC shortly after leveling off at 7000 ft. At that time I noticed that my primary (analog display) cylinder head temperature (CHT) gauge was indicating well beyond the redline; it has indicated normally until this point.The aircraft is additionally equipped with a JPI EDM-700 engine analyzer; which provides a digital readout of CHTs on all six cylinders. This indicated normally; as did all other sensors and the performance of the airplane; and so I suspected a failure of the primary CHT gauge or its sensor.However; I thought it was safest to terminate the flight and land. I considered landing at ZZZ2 but decided on ZZZ as its further distance would permit a more normal descent; and terrain and mountain obscuration around ZZZ2 seemed higher risk.The ZZZ Center Controller appeared to be working several sectors; as I could hear her but not the other aircraft. I tried twice to get her attention with a 'ZZZ Center; Bonanza Tail Number' transmission; looking to initiate an discussion about my need to return to ZZZ. Both calls were unanswered and the controller continued addressing other aircraft with routine duties such as providing altimeter settings to aircraft checking in. At that point; I [requested priority handling] and request direct ZZZ; explained that I had a CHT overheat indication; engine failure was possible but everything was running normally at this time. The reply was immediate and I was cleared direct ZZZ and given a series of descents from ZZZ Center and; later; ZZZ Approach.The descent was normal; and when I broke out of IMC I was cleared for a visual approach to Runway XX. I was still quite high and advised ZZZ Tower that I planned to circle on downwind before entering a normal pattern; which was approved. The CHT indication returned to normal shortly after landing; and the aircraft's A&P mechanic was consulted.Reflecting back to improve safety; I believe that diversion was the right decisions to ensure a safe outcome of the flight; but think that I could have been more patient in waiting for a reply from ZZZ Center. I did not initially intend to [request priority handling]; but ultimately did so after my initial calls to ATC did not get a response. In hindsight; a PAN-PAN call probably would have afforded the same attention while reducing everyone's stress level (including my own).