If the light begins to flash after the bulb check, it indicates that someone used an invalid key fob to start the engine. So don't be mixing up your FOBs between vehicles. The engine may be shut off after two seconds.
SECURITY INDICATOR FLASHES FOLLOWING BULB TEST - Invalid FOBIK in ignition switch or Key-related fault
SECURITY INDICATOR LIGHTS SOLID FOLLOWING BULB TEST - SKIS system malfunction/fault detected or SKIS system ineffective.
You will need a ODB scanner that can read body modules to figure out whats going on.
For those equipped with the security alarm, the vehicle security light in the instrument cluster will flash when a security alarm is triggered. Unlocking the doors with the FOB, Passive Entry, or turning the ignition on will reset these. If not, there probably is a fault somewhere.
When the VTA begins the passive or active arming sequence, the security indicator will flash rapidly for about 16 seconds. This indicates that VTA arming is in progress. If the status of the ignition switch transitions to ON, if a door or the liftgate is unlocked and opened by any means, or if the RKE PANIC button (if equipped) is depressed during the 16 second arming process, the security indicator will stop flashing and the arming process will abort. Once the arming sequence is successfully completed, the security indicator will flash at a slower rate, indicating that the VTA is armed.
So perhaps the doors being locked starts the VTA arming, but something else is preventing remote start. Are these OEM FOBs?