Ok, if the Navcore works with the remote after installing it then there must be something the TT changes to wreck the remote.
No hardware failure but a software problem.
You dealer can't do anything about this and a diagnosis won't give you an answer - their systems won't check more on the TT than what you get by checking the information page on the device itself.
I am not familiar with the things Home puts on the SD for an update, but as long as the same Navcore is re-installed all settings will be kept and same for the files on the internal memory that don't come with the Navcore, like favourites, settings and so on.
It is just a shot into the blue but maybe....
Make a SD to get the remote working.
Change one or two of your favourites, e.g. rename or delete.
Change some settings in the menu, e.g. different voice, change menu language.
Turn the Carminat off the normal way and check if after the start the remote will still work.
Both the settings file for the map and the Navcore will be changed and hopefully one of them also contains the settings for the remote.
If the above does not help, try again and in the menu pair another BT device, if your phone is already connected remove that from the carminat and pair a different device.
This will change the pairing information and if corrupted (the original) it should keep the settings for the remote.
Not 100% if you can simply remove the remote from there too so you can pair it fresh, depends if you have to confirm something on the carminat for the pairing.
If none works I am seriously at the end of wisdom