Pilot on training flight with Instructor reported suspected engine fire in flight.

Narrative

Prior to the incident; we had conducted two laps in the pattern with normal operation We were asked to land and taxi back due to traffic waiting to depart the airport. While waiting to depart for another lap in the pattern in sequence with other aircraft; we noticed a 'campfire' like smell that was intermittent. We opened windows and doors to identify source of smell and the smell dissipated. We did not see smoke when the instructor opened the door and stuck her head out to look at the exhaust. We concluded that the smell originated from an external source and continued the flight. We were holding short for several minutes with nominal engine operation. On the runway; full power was applied; verified engine instruments were green and takeoff power was normal and takeoff approximately 1500 feet down Runway XX. Per track log from Sentry/Foreflight; at 200ft AGL we experienced a drop in power and performance was not as expected. The airplane was climbing slowly and climb angle was shallower than normal to maintain adequate airspeed. The passenger (commercially rated) and instructor noted engine noise and vibration change. We turned crosswind early at 715ft AGL and kept the downwind within gliding distance. Midfield downwind the instructor noted excessive heat on shoes; legs; and in the cabin plastics and side pocket. Oil temperature and pressure were normal at this time; but EGT gauge was inop so no reading could be taken to understand state of exhaust gas temperature. EGT was squawked INOP as of a month ago and placarded prior to the flight. At this time; cabin plastics and air vents were hot to touch. I requested a short approach and full stop landing; and touched down on the 1000' markers and exited the runway immediately at [Taxiway] 1. Instructor pulled mixture and I pulled fuel cutoff; the passenger exited the aircraft first followed by the instructor and myself. My flight bag was in the baggage compartment to make room for our passenger; so prior to the flight I had put my handheld radio in the pilot side pocket in case it would be needed in flight. I called the Tower on my handheld after exiting the aircraft to notify that we evacuated the airplane due to a suspected engine fire. Fire department (FD) was called and noted that excessive temperatures were registered in the right hand cylinders about 15 minutes after landing. On exit a post-flight walk around; we also noticed that the exhaust stack was charred black which was not present before the flight. Airplane was towed back to the ramp after the FD called all-clear on fire risk. Due to the proximity to the ground and runway and need secure the aircraft prior to landing; and emergency was not declared on frequency with the Tower. There were no injuries and no external damage to the aircraft was visible on the post-flight inspection with the exception of the charred exhaust stack.