Thanks for your replies.
I did some tests today, including the one that ice9mike did (swapping the connectors between 5th and 6th string). This made the problem on the low E go away completely and to some extend it was audible on the A.
This proves indeed that the problem is after the piezos.
I also took the A string off and put a .042 in place and tuned it to E, so I had two the same E's as the 5th and 6th string. 6th obviously had the chirp/pling problem, 5th not. When I switched the connector on the controller board around, the problem stayed on the 6th, but much less. 5th had no real problem.
This is contradictive because it does indicate a faulty piezo.
Maybe it's a combination of things.
In all cases the piezo on the 6th saddle gives less output than the others.
I was physically going to swap saddles, but since I'm only owning the guitar for a day, I'm going to return it and test another one. The only thing I'm a bit afraid of is the setup of the guitar. I tested 3 JTV-89's on Saturday in the store. Only the one I eventually took had a really great action and overall feel. I hope I can get the same feel on the new one I might be getting.
I will let you know what happens.