I recently bought a second-hand Brava SX 16V but did not get the radio code. Is it possible to connect a CD-changer without disconnecting the radio from the battery? If not, how can I find out the code?

You could get the code from fiat dealer don't know how much some charge lots some do it for free. since you are in scotland you may have to go to arnold clark from what i've heard about them they will charge you just to speak to them.
most car audio shops will recode your raido for about 10 15 quid, if you are buying a new cd changer see if the shop wherer you buy it will re-code your radio for free as you are giving them money anyway.

Sorry for got to say, Welcome to the fiat forum.