C172 Flight Instructor reported cockpit smoke and fumes during climb. Diverted and landed uneventfully.

Narrative

This occurred shortly after departing from ZZZ1 on VFR flight following to ZZZ2 while flying through the ZZZ3 Class C just north of ZZZ. During a routine training flight in the climb phase of flight; I smelled burning rubber inside the airplane. Additionally; the AVIONICS BUS 2 failed; and the Multi-Function Display and the associated Garmin Line Replaceable Units (including Com/Nav 2; the audio panel; ADS-B; and transponder) failed. With a high workload dealing with an emergency; a student; and with the ZZZ TRACON frequency being congested; I opted to tune to ZZZ Tower for help instead. Unable to use the copilot's side headset or mic; I used the pilots side headset jack and push-to-talk button. I checked in with ZZZ ATCT and [requested priority handling]; and I was cleared to land on Runway XXL at ZZZ. Upon landing; I shut down the aircraft on a taxiway; safely evacuated and called the ZZZ ATCT over the phone for help. After towing the aircraft to a local maintenance provider; I called the NTSB and spoke with their Duty Officer about reporting an incident. After providing the necessary information; the Officer said that he would not be pursuing the case and nor filing an incident.The local maintenance provider discovered the AVIONICS 2 switch showed signs of a short circuit. The metal contacts inside the switch fused themselves in the ON position. The STANDBY battery was also found to have died during this post landing inspection; but indications in the preflight and run up were nominal. No thermal circuit breakers tripped.