Normally 240 vac connect pumps in U.S. use a double pole, single throw relay. There should be no neutral connection for a 240 vac pump. There should be a green grounding wire to case.
The reason for using double pole is to avoid leaving 120 vac on motor winding to ground.
Solid state switches have about 1.6 vdc drop due to two equivalent diode drops associated with triac switch. What is more important is the run current. This causes the steady state heating of the SS switch. The SS switch needs to be heat sunk sufficient to dissipate the heat generated by its voltage drop (1.6 vdc) times the current it is passing.
You can place two SS switches, one on each L1 and L2 lines, tieing the control lines together.
You have a wiring problem and may have already blown out the triac. When you said you switched Neutral line that's a red flag that you may have screwed up the connections, as neutral should not exist running to the 240 vac pump. (neutral is white wire, L1 & L2 should be red, black, and ground wires should be green.).