"But the sense of injustice we feel makes me wonder what other courses of action could I explore?"

Reduce the speed limit warning threshold perhaps?

36 in 30 is exceeding the limit. Perhaps set the threshold at 33mph.

The warnings have been set wrongly if the OP is relying on them to avoid prosecution - the APCO/NPCC guideline is that limit +10% + 1mph isn't prosecuted but limit + 10% +2 mph is prosecuted - so the OP needs to set the warning for 34 mph because 35 or 36 is too high.

None of the equipment in any normal car is calibrated accurately enough to be used in court - you'd have to show that the equipment used by the police was faulty - so in practice, no chance and the penalty likely to be higher if you challenge it.

Accept the speed awareness course if offered and also learn that your car technology isn't perfect.

It was cameras. Had there been an interaction with a real policeman/woman I am sure there would have leniency shown. We literally were the only car on that road and, ironically, we were passing the police county headquarters. Oh, I should mention the certificate of conformity for the camera expired some ten days after we went through it.

If the car's system works by using a camera, it may be that the speed limit sign was not visible and therefore the fine is not enforceable (unless there was road-lighting).

Oh if only it were that simple! So you rock up at court, say that the camera used by your driver’s aid did not capture the “30” sign, so it could not have been visible, therefore you could not have seen it so the limit is unenforceable. Case dismissed. Job done (or possibly not).

Back in the real world, the measuring equipment used by the police is deemed to be reliable provided it is an approved device and was operated in the correct manner. The police will undoubtedly be able to prove that. The burden then shifts to the defendant to cast doubt on that reliability. All you have is that your unapproved and uncalibrated driver’s aid did not warn you that you were exceeding the speed limit. Put yourself in the position of the Magistrates: which version of events would you prefer?

Had there been an interaction with a real policeman/woman I am sure there would have leniency shown.

I fear you may have been disappointed. The “leniency” shown is the police do not usually enforce a speed limit until a speed of (Limit + 10% + 2mph) is reached. There is no justification to provide leniency over and above that

You should be offered a course for that speed (which is further leniency as the police have every right to prosecute you through the courts).

Setting the speed warning to 30mph in a 30 limit is probably a better idea than setting it to 36.

I'm a bit bewildered. Have you not got a speedometer?