I can honestly say 100% confidently that on a 147 the rear sidelight not working or staying on is a fault with the body computer. This is a pain in the **** and will certainly need a new one. not cheap at all £350 for the "computer" which is actually a circuit board and situated behind the fuse pannel on the drivers side.
This is an Alfa only part, the unit cost is above but i would recommend they fit a new unit as they are delicate little buggers. total cost around £500 for the unit and fitting.
many people will have little tricks to get it working, one of which is piggy backing the wire from the other side but this wont get rid of the failure light on the dash.
I would suggest if you are going to piggy back the wires then get an auto electrician to do it, this will cost about £150 but if you know a mate or are confident then it will work out better cost wise.
Resetting the ECU does not work for this, diagnostics wont do anything with this either.
Whatever methods is used to fix the problem is only temporary, if you piggy back the wire you will soon get fed up of the failure light on the dash, and once the body computer has an issue, other things may stop working to, within time not straight away.
You could try and see if you can get the board repaired but trying to find someone to do it is near impossible and not guaranteed.
Sorry this isnt the news you wanted to here but that will be the fault.
WOOF BARK DONKEY!