Why not do away with the SSR/IGBT combo and use an NPN transistor driving a P-channel mosfet instead? See attached.
When choosing the mosfet, bear in mind that the resistance of a filament bulb when cold is much lower then when it is hot, so the inrush current is correspondingly higher than the steady-state current - about 15x higher, maybe even more.
Running the Nano direct from 13.8v will not harm it, the main thing is to avoid drawing too much current from the +5v rail in order to avoid overheating the regulator.
