Protecting Solid State Relays from Short Circuits
You need to monitor the current using a low resistance in series with the load (bulb). Or possibly, the drop across the relay can be used to calculate the amount of current taken by the load.

Once the drop exceeds a threshold, the relay should be turned off.
Make sure that the detection mechanism and relay is fast enough to allow for abrupt termination.

Otherwise, go for a fast blowing fuse in a holder; solid state stuff has low thermal mass.
Replace the fuse while replacing the bulb.