Yogurt on a PID controller
The real problem seems to be your mechanical setup, not the PID control detail.  PID control may not even be what you want at all.

I'd start by putting the container of yogurt culture inside a water bath, then attempt to regulate the water bath temperature.  The extra thermal mass of the water will lengthen the dominant time constant, and won't depend on what is going on inside the container much.

With a longer time constant, simple thermostat on/off temperature control should work well enough.  You want the heater power to be low enough so that the water bath only changes a little, like maybe 2-4 degrees, within one time constant.  Simple threshold-detecting on/off control will then work very well.

Instead of adding hysteresis to prevent rapid oscillations at the setpoint, just hold the output fixed for a short time, like 1/50 of the time constant or so.

A long time ago, I made a temperature controlled water bath for photographic film processing.  The mechanical setup was one of those plastic "Rubbermade" style tubs, a fish tank pump to keep the water circulating, and a 300 W off the shelf immersion heater meant for a coffee cup.  Two thermistors in series placed on opposite sides of the tub provided the temperature feedback.  This was before the age of microcontrollers, so a flip-flop sampled and held the on/off signal every 64 line cycles.  That controlled a relay, which switched the immersion heater on and off.  It worked really well.  Once it got to regulation, the temperature stayed fixed to a fraction of a degree F.

You are over-thinking this with a PID controller.