You will have to figure out how the new harness "works." Surely must have a "crank" wire from the ignition switch, so this goes to EITHER of the "push on" terminals on your start relay.
Now, to make the start relay actuate during crank, you must GROUND the remaining push-on terminal through the neutral safety switch.
The harness for the newer NSS/ backup switch comes up over the top of the trans, and splits off at the bulkhead connector, where the two backup switch wires feed through the bulkhead, and the NSS (center terminal) splits off and goes over to the remaining push-on terminal on the start relay.
THE BIG thing that is probably missing in your GM/ Mopar lashup, is you must provide a wire from the ignition switch from the old "ign1" (traditionally brown) resistor bypass circuit.
So you options on that score are......
If you are running a Mopar ignition switch, just run a new wire
If you are wiring an older car (54 pickup, etc) and running a non-Mopar ign switch, you can either
use a newer "Jeep" start relay which has an extra set of contacts
use a separate "bosch" relay actuated by the start relay
use a great big series diode from the crank signal (yellow) wire
The output of any of these options goes to the coil side of your ballast, for a "hot" start voltage during crank.
the backup switch section of the NSS (two outer contacts) will feed through your builhead. One must go to whatever is provided in your new harness for fused switched backup lights, and the other must end up going to the rear to feed the lamps.
Newer Jeep relay: