Chevy Cavalier 1998 180000km in New Zealand

Hi, I am a Lcoksmith and we are having issues with a Cavalier (re-badged as a Toyota in New Zealand) the car had a damaged ignition switch, we had to die grind the locking pin to remove barrel, once barrel was out we sourced another barrel, rekeyed to the original key etc. Then we found a small magnet like transponder device in the inner barrel and a receiver device on the barrel housing, I tried to remove the small magnet however it was damaged (very brittle and probably not made to be removed).
The small piece of magnet became very fragmented but I was able to get it into the new barrel, however the theft system is enabled and will not allow the car to start (car will start for 2 or 3 seconds and then stop)
Any suggestions? Can these be bypassed?
These cars are not common in New Zealand and it has been difficult to find information.

Many thanks