Hi Soren (@sorenszabo),
I'm afraid you may be conflating two separate issues here. The annoying alert as you mention is presumably the one that says the phone can't tell if the screen is a genuine Apple part or not. Starting with the iPhone 11, Apple started pairing screens to the logic board and that warning is what happens when it sees a screen that isn't paired to it. Since only Apple can do that pairing, anyone who isn't Apple that replaces a screen will get that warning. It's my understanding that the pop-up warning will go away after a couple of weeks or so, but the badge on the setup app is permanent.
Technically there is a way to get rid of that warning, which is to move the touch module IC from the original display to the replacement. It's a very difficult and delicate operation and probably isn't worth it, to be honest. Outside of that, paying Apple to replace the screen is the only other way to get rid of that warning; screen replacement on the 12 models is available on the self repair program, but you still have to buy the screen from Apple for full price and rent their equipment to do the job.
Now the True Tone function is an entirely separate issue. Fixing True Tone will not get rid of the warning, and getting rid of the warning will not enable True Tone. I have not worked on later model phones past the iPhone X with regard to that so I can't address what Apple may have done on those later models to make it harder to copy the True Tone data.
There is precedent for that, unfortunately. As of the iPhone 11, Apple changed the way the battery health was stored. As of that model it is no longer possible to update that information on the battery's BMS board so you can no longer reset the health percentage or the cycle count. Aftermarket manufacturers have responded by creating a tiny circuit board that gets inserted between the battery and the logic board that allows that information to be modified again. It appears they've done something similar with the True Tone data (aka the MtSN number) on these newer models, but I haven't done enough research to know exactly how it works or more importantly how to get around Apple's new restrictions.