I've never used a heated mixing bowl, but I do sous vide, which I believe works on the same principle as what you're trying (namely that you can't overcook it because the cooking temp is where you want the final internal temp of the food to be).

Anova has a pretty popular recipe for creme brulee which specifies 176ºF (80ºC) as the correct temp. They also have an older, less popular recipe that specifies 180ºF (82ºC). This recipe for traditional baked creme brulee says to bake until the internal temp is 170F-175F, but with residual heat I would expect it to continue cooking until somewhere between 176-180F once out of the oven and water bath.

If I were you I would set the mixing bowl temp to 176ºF (80ºC) and see how that sets up. If the final consistency isn't quite where you like it, you can try it a little higher next time.

I was finally able to get back to this and make some edits.

My findings:

My first assumption was that egg yolks would behave like cornstarch and that was wrong.
Cornstarch, once it has reached its "coagulating"/"setting"/whatever temperature (assuming uniform temperature distribution and no evaporation) will set and further heating it or increasing the temperature will not make it "set" any harder.

Egg yolks are different in that regard. Egg yolks (and eggs in general) will begin to set/coagulate from a certain temperature and will set/coagulate/harden to a different extent at different temperatures.

Furthermore, in my observations, creme brulee behaves differently than cornstarch-based recipes (like chocolate pudding, for example).
In the case of cornstarch chocolate pudding, once it reaches the required temperature for the cornstarch to set (about 95 degrees Celsius) the cornstarch sets and the mixture becomes visibly much thicker (remember you always stir pudding), which is a clear giveaway sign it is ready or almost ready to be taken off the stove (or out of the mixing bowl in my case). Furthermore refrigeration does not drastically change the cornstarch pudding thickness.
But creme brulee will remain quite runny even after the yolks have started to set (at least while it's still being stirred as to not burn) or have reached maximum "setness" or "coagulation" for a given temperature. Mine actually looked like a cup of melted butter up until I put it in the refrigerator. Which brings me to my next point - refrigeration drastically alters creme brulee thickness.
(The above is not true (which is why I striked it through), creme brulee will visibly thicken and set even more than cornstarch pudding before refrigeration.)

What I tried:

I tried performing an experiment where I would slowly and gradually heat egg yolks to certain temperatures and observe what happens to them both when stirred and when not but I was unable using my equipment.

Firstly heating egg yolks without stirring them causes them to develop a "cream" - a sort of surface membrane that is a bit harder and traps heat inside, like milk does when you heat it up without stirring. That and them getting heated non-uniformly due to the parts that are in contact with the hotplate coagulating first ruined that approach.

The only observation I was able to make thus far was when I was also stirring them but that also failed due to my equipment being made with bigger portions in mind and the temperature control isn't that good for a couple of egg yolks.

The reason for the runniness and my general failure to conceive Creme Brulee was that I was always stirring it, never leaving it to set properly and, furthermore, my heated mixing bowl isn't made with that in mind and leaving it to set didn't go too well, either.
(You can make creme brulee just fine whilst stirring it, my problem stems from elsewhere.)

Findings are based on this recipe:

• 6 egg yolks
• 500ml cream
• 300ml sugar

Conclusion:

The problem in my case was that I was stirring the mixture non-stop, which prevented it from setting at desired temperatures, instead allowing it to set in a very unappealing way only at high temperatures. Unfortunately my equipment does not allow me to make creme brulee, like I had hoped, and thus cannot comment on the proper temperature. The ~85 degrees Celsius looks good, though.
So far I'm still on it but I'm guessing the problem is in my hotplate, not the stirring (for example when making creme brulee on the stove you need to stir it very, very fast).