Hello,
I believe there is damage to the central locking control module, which will require testing.
This is no easy module to test physically, instead, it should be checked with a professional scan tool.
The big issue here is that the connections to the module will look like a pot of spaghetti noodles on steroids, and some of those wires are analog signals and some are digital.
Testing with a scanner can be done pretty quickly, maybe an hour or two at the most (for a good VW/Audi/Porsche shop), but testing bit by bit... you may be done by May if you quit your job.
Sorry, I'm just trying to give it to you as straight as possible. I'm a VW/Audi guy, and I've been through these.
The central locking control module sends data to the instrument cluster, has interior light functions, is the power window/lock/trunk/gas tank/etc.. controller, and does pretty much... everything else including anti-theft functions.
On any american car it would be called a body control module.
Seek the help of an independent VW/Audi or European auto shop. They will have done this repair before.
So would replacing the whole central locking control module fix this problem? Because I have found one on eBay fairly cheep and if you are positive that that is the problem I want to buy it.
If that is the issue, yes. But you really should have it diagnosed with a scanner.