Do you connect the PID such that the controller sets the current to a certain level, or do you connect the PID such that the controller adds/subtracts current to the instantaneous current to reach the desired level?
There are different PID algorithms that do either of those two things. The latter type is called a velocity algorithm (because it controls the velocity of a hypothetical mechanical valve motion rather than the position of the valve). Since the valve performs the integration there were once some advantages to this method- for one thing it automatically stops integrating when the actuator hits the limit, which helps with overshoot due to integrator windup. However, if you are using derivative control it must be double differentiated- which tends to make that contribution very noisy.
By the way, controlling the current through the resistor is usually not a good idea. What you want to do (typically) is to control the power. The resulting nonlinearity from voltage control will usually make the controller hard to tune well over a range of setpoints (it will either be too sluggish or not stable enough).
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.
Best Answer
The relay output is SPST rated at 5A:
You cannot use this output to directly control a typical DC-input SSR without an external DC power supply of some type. It's just a normally-open contact.
You should not switch anything like 5A with it either, unless you want it to last a very short time. Typically life at full rated current is only perhaps 100,000 operations, which might be weeks or months 24/7.
They should repeat the datasheet specs within the manual, but I imagine this is just some rebranded no-name Asian product anyway. I doubt Love makes this kind of control these days (they did, many years ago).