I haven't used a pic-kit3 programmer, and it may not make any difference, bit how long is the connection between the programmer and the device? It should be as short as possible, ideally not more than 100-150mm. Having a hunt around with an oscilloscope would be the next thing.
Ok, I tried to shorten the connectors but the problems is still the same. I don't have an oscilloscope and I think that would be a bit too much for this project. At least for now.