B737 NG Captain reported landing safely after shutting down the number 2 engine as a precaution following the illumination of the filter bypass light and increased oil temperature.

Narrative

We were flying north from ZZZ just south of ZZZ1. The number 2 engine filter bypass light illuminated. The First Officer was the PF; so I referenced and actioned the QRH. The filter bypass light checklist is very short; and as we would be reducing thrust in an engine and possibly shutting down; I immediately requested a descent to FL310. After retarding the thrust; the bypass light became intermittent at varying amounts of thrust. I then notified Dispatch and coordinated with my Dispatcher and Maintenance. Maintenance advised me to follow the QRH; but with an intermittent light gave me oil pressure and temperature parameters to monitor. After descending to FL 270 and being cleared to descend via the ZZZZZ the bypass light became steady and I noted an increase in oil temperature we elected to [request priority handling] and shut down the engine per the QRH Engine Shutdown checklist. The checklist calls for landing at the nearest suitable airport. As we were already cleared to descend and had briefed the arrival into ZZZ3 we decided that ZZZ3 would be the best option for landing and would prevent the introduction of new threats inherent in a diversion. I briefed the flight attendants with no special instructions and no further issues expected after landing. We descended via the ZZZZZ arrival; I took over the flying duties and landed uneventfully on Runway XXR in ZZZ3. We taxied to Gate and deplaned normally. Maintenance write up was completed and Chief Pilot called for debrief.