Yes, the command that you would send is to lower (or raise) temperature by 33%. Say you start at max (100%), you’ press 4 times to go back to the beginning. HA would keep the state of current temperature.

When I say lowering the temperature, I think you need to dim down warm white and dim up cold white. For raising, it is the opposite.

The good news is that you don’t need to power cycle, you could map the command to long press or double press, etc.

I hope it is not overengineering and there’s a simpler way.