Arduino to PID control a fridge?

I have a setup to control the temperature of a fridge (used for fermenting beer !) with an Arduino. Basically the arduino reads the fridge temp and apply cooling/heating via SSR’s connected to the fridge compressor and a light bulb. I use a target temperature and “dead band” approach for ths logic.
But I would like to improve on the accuracy of the fridge temperature by implementing PID control. The problem is that the cooling (compressor) and heating (light bulb) is either on or off, I cannot vary the power applied. Also the compressor cannot be turned on/off too quickly since that might not be good for it.

Does anybody have any good ideas on how to approach this? The temperature could be measured in a big bucket of fluid (beer :slight_smile: ) to slow the system down…

Would it be possible to use Arduinos PWM output for this with large timeperiods? Minutes, or even hours ?

heating (light bulb) is either on or off

Will you be changing this to PWM?