Detecting a short in an enclosed circuit with a PLC
Your test method should work if you have a current-limited supply for the 12V. The supply provides 12V up to the maximum possible current for a working device, and limits above that. Many 12V switching supplies would probably work, but you might want to use a lab supply with programmable voltage and current and measurement . Some might be able to reverse output voltage in a programmable manner, or you might need to add relays. The one I linked can be interfaced with the PLC via Ethernet and uses standard SCPI commands, but there are many other options. 

If you want good measurement of the leakage/shunt you'd probably want to add another LAN interfaced bench meter in series  with the output so the current could be measured to uA or below. 

This would be a pretty standard type of test setup (a few rack-mount devices wired to a test fixture with some operator interface). 

Of course a bespoke custom PCB design could replace the off-the-shelf test instruments at considerably less parts cost, and that would be the first impulse of many engineers, but by the time you're done paying engineering time and maintenance you might be better off buying the calibrated traceable instruments. 

The reverse LED leakage of all the LED strings in parallel would limit how accurately you could measure the shunt resistance "short". It's also possible to measure in the forward direction with a relatively low voltage but I think reverse would be more accurate at a given test voltage. Typically, at room temperature, that leakage is very low for most LEDs, but you'd have to check the data sheets and maybe make some measurements. It would be best to do the reverse test first while the LEDs are cold. 

So the PLC or computer would need a program that would configure the one or two test instruments for each phase of the test plan, read the results for each and then present a summary to the operator. Maybe with some additional operator interface buttons, lights, beepers or whatever.