I think the same relay also powers the footwell courtesy lamps, i know it does in my 02 PT, i haven't checked the service manual on the KJ, but i bet its the same as usually all the interior courtesy lights are on the same circuit.
Must be a bad connection somewhere. The map lights have fused B+ at the connector and the ground is switched by the relay, i learned this mapping out the wiring to swap out for an EVIC as the service manual for the 03' shows the fused B+ to the lights are the same as the fused B+ to the EVIC and the chunk of harness that came with my EVIC has a factory splice to feed both the lights and the EVIC off the same B+ line(hint for those swapping to an OTIS or EVIC, you don't need to run a new wire for the B+ line).
So if you can turn your lights on and off by pushing them but not with the doors then there must be a bad connection in the switched ground from the lights to the relay behind the fuse panel.
Try pulling down the map light assembly and disconnect the plug, use a test light or voltmeter and one should have power, the other is the switched ground. To test it put the negative lead from your test light or voltmeter on the switched ground, and the other to the power in that plug and then open and close your door, you should see that ground get switched on and off and your test light come on or your voltmeter show voltage there. If not then the problem is farther down the line, if it does then you have a bad connection at the light.
I will see if i can find the wiring diagram in the service manual.