One of your main stated issues is that your supply water temperature has a “swing” that results uneven output.
So you could monitor the supply temperature, pressure and implement a Arduino switching control of the power to the shower head without doing anything inside the shower enclosure.
Yes you would need to experiment a little with what switching frequency is needed for the desired output, but once you dial in the values you can have a very controllable output based solely on data from the cold water supply.
In practice you would turn your shower head on high, the Arduino would read the current water temperature and pressure and pulse the power to the shower head so that the temperature is lowered.
You should note that SSR often fail closed (full on), This could lead to over heating the water. Your circuit should include a NC relay inline with the SSR to break the circuit in the event your code detects an over temperature.
Might be a good time to invest in a safety switch although many of these direct heaters already incorporate them within the control, cannot see much like that in the photo.
The additional one would be to cover the "additions" planned.
Have a feeling it will be needed.
10Kw on 220v would be 45a so a 40A breaker as quoted would not work for long.
Neither will 6square mm cable if over more than a few metres.(voltage drop @45A)
I can work it out for you if you have the need.
This looks like an interesting project. Where is the heating element? Is it inside the fixture shown in your photo?
What does the adjustment wand actually do? Is it an electrical switch or a bi-metal adjustment? If it's electrical, it would help to see a schematic of the current control circuitry.
Not a lot of detail to scrounge any real data but did notice it is 7.5Kw not 10kw.
Hense the 6mm cable size and 40 A breaker quoted by the Op, still that is dependant on final sub-circuit run length.
Looks to me like one of those things one would want to be steering clear of entirely.
Grounding...well it apparently has one, maybe.
Who knows what one can make of the product description.
IP24 ...mmm... first digit ..... Protected from touch by fingers and objects greater than 12 millimeters.
second digit... Protected from water spray from any direction.