Jeff, here's some more input what I found out.

As said before I tried to change the woofer drivers. But this did not make any difference.
Also I changed the tweeter driver for another one. But this also made no difference. 

Now I noticed something strange. If I take out the tweeter driver, without replacing it with another tweeter driver, the normal bass sound is back and the shrill sound is gone.
And as soon as I connect the tweeter driver, the bass sound is gone again and the shrill sound is back.

What do you make of this??? 

The only thing that pops into my mind is that the tweeter level is way off or it's got some distortion. Since you've swapped tweeters it'd have to be in the crossover or amp. I'd shove something like a 5 KHz or such signal into the speakers, something up in the tweeter range, a sine wave, and measure the voltage across the tweeter on the good speaker and the bad one to see if they were different. If they're the same, something really weird is going on in the electronics I can't fathom.

You do seem to have the most interesting problems! 🙁

Thnx Jeff! I'm not sure if I like to have the most interesting problems ... but hey, let's call it a challenge. ;-)
The BeoLab just is a too beautiful product to put it aside.

Thanks also for your new suggestions Jeff. I will try to put a 5Khz signal into to speakers.

I must say, I'm a bit dazzled by the fact that the bass is back in the 2 lower speakers and the shrill sound is gone, when I take the tweeter driver out.
But that the bass is gone and the shrill sound back when the tweeter driver is connected again.

Sounds like the physical tweeter driver connection is triggering both the bass to disappear and the shrill sound to appear.

??? 

I suspect something is going on in the electronics that's not easily to diagnose without pulling the whole electronics package and putting it on the bench. I think, sadly, you've about exhausted what you can look at yourself and perhaps it's time to throw it at a repair tech.

The best you can do I think is, since you have one working speaker, is to measure and test as many things as you can where you can compre one speaker to the other, and hope for luck in finding something different that you can trace to a fault. Not always easy though.

And yes, they are too beautiful and iconic to abandon. I listen to mine every day.

Did you manage to fix your speakers? Would be curious to know what the issue was?