Shower: Can this single-lever mixer be converted to a thermostat?

In the relatively new house (4 years old) that we have rented, unfortunately only a single-lever mixer is installed in the shower. It has actually been annoying me from the very beginning, since I installed a rain shower, but even more so because you often have to adjust the amount of water and the temperature always changes in the process (really annoying with children). Therefore, a thermostat would be much more practical.

Now I only ever see these thermostats in one version, where water is supplied from two sides. But here we have a single-lever mixer without visible connections. So my question is: Is it possible to convert this to a thermostat, and if so, is that also possible in a rental house, or do you have to make irreversible changes for that?

Many thanks in advance for all ideas on this topic :)

Are there any details I need to add to this so that someone can say something about it, or is it simply not possible?

This is a concealed fitting, the manufacturer should be visible when the lever is removed, but the fluctuations while showering are not normal. Probably something cheap installed.

A replacement is very difficult because it is concealed.

Write something about the manufacturer.

If the amount of water is not changed, then the temperature stays the same and there are no fluctuations. Only if the amount is reduced, for example, then the temperature also becomes higher, but it remains constant.

Ok, if an exchange is practically not possible anyway because of the in-wall issue, then I think I'll save myself the effort of dismantling :/

Behind it is a built-in unit. Replacement with a thermostat is generally not easily possible.

I'll just jump in... more out of curiosity, since I've thought about it before....

Someone once told me (although not a professional) that a later retrofit is possible for some. For example, I have a Hansabluebox with a shut-off. But I dropped the idea again because it wasn't worth a few hundred euros to me to replace an almost new faucet.

- Is that fundamentally possible with the Hansabluebox?
- How does that work then? I need both water control (with a diverter for overhead shower/hand shower) and the actual temperature control on one faucet. Traditionally, these are separate.