I just had a thought when reading your "click" comment - Are you positive you have the right PID? They can be ordered in relay output control ( usually designated with an "R" in the model code) or with SSR output control (Code S). If you connect a relay type PID to an SSR with no DC power in the circuit, nothing will happen when the relay "clicks". I see that the one you linked to is dual output, but there is a possibility that it is wrong, or the SSR out isn't working. Double check that the program is set for SSR out, and that there is voltage coming from the SSR +/- terminals. See last paragraph for how to fix that. Some PIDs have to be programmed for either R or S control. Make sure you are in the SSR mode if that applies.
Earlier thoughts:
Are the wires from the PID to the SSR connected to the "NO relay" terminals or the SSR +/- contacts? The SSR needs 3-230VDC to operate. This applies to PID units with both SSR and relay output.
Did you do the test where you apply a DC voltage to the SSR? That test will tell you if the coils power circuit is right. If applying DC directly to the SSR from an outside source causes the coils to heat, the problem is either that the PID is wired wrong to the SSR, or the PID isn't working right.
Here is how I would troubleshoot this problem:
1) Temporarily put the power wires that are on SSR terminals 1 and 2 on the same terminal (say, both on #1). This should make the coils heat up when you flip on the main power switch. If the coils don't heat up, there is a problem with the coils wiring. If the coils heat up, proceed to next test.
2) Put the power wires back on SSR#1 as normal. Apply 12VDC to the SSR terminals #3, observing polarity. Switch on the main power switch. The coils should heat up. If they don't heat up, the SSR is bad. If they heat up, proceed to next test.
3) Check the SSR +/- voltage from the PID while it is in the RUN mode. There should be 9-12VDC on the two terminals. If no voltage is found, either the PID is bad, the PID is not an SSR type ( or programed for SSR), or the PID isn't in the RUN mode.
If you are pretty sure that the PID is trying to turn the SSR on and off (relay clicks), but there is no power from the SSR +/- terminals, try this:
Use the DC source you tested the SSR with in the above tests and connect the + to the SSR ( should be + terminal #3), the - to one of the PID relay NO contacts, and a wire from the other PID relay NO contact to SSR - terminal ( Should be - terminal #4). With this configuration, you are running the SSR on the relay out circuit, not the voltage out circuit. If it runs the coils this way, the PID has a problem with its DC out circuit, but you can run it on the relay circuit just fine.