With the diode, only the upper or lower half wave conducts and that can alter the proper functioning of the SSR.
You could try connecting a resistive load in parallel with a diode + vibrator, for example a tin soldering iron or an incandescent bulb of about 100W. Maybe then it will work
Also try reversing the cables at the SSR output terminals.
If it were in parallel something would have burned, at least the fuse.
Yeah, my bad....was thinking a back EMF diode on a DC circuit.