System that does not work
You are not supplying enough current to the SSR to turn it on.

According to the datasheet the trigger uses 7.5 mA @ 12 V.

5V is probably enough, but your level shifter does not allow enough current.

You can probably get this to work with just a logic-level MOSFET and 5 V:







simulate this circuit &ndash; Schematic created using CircuitLab