One switch, two different functions, two different troubleshooting problems. With the obvious stated, I can offer a few things you can check (separated by system). First make sure the switch is adjusted properly on the trans!
Since you didn't mention any work yet on the neutral switch we'll go there first. Starting at the neutral switch there is a double wire that is red with a blue stripe (both wires). This is the neutral safety circuit and terminates in a double pin plug (one male, one female), at the firewall where it plugs into an identical plug with it's wiring passing into the interior (same color wires). Once inside the wires are separated with a single connector on each, one male, one female, under the dash near the ignition switch. Find that connection. They should each be plugged into single wires. One of those will go to the main harness firewall plug and out the stater solenoid. The other goes to the back of the ignition switch. These two wires CAN be plugged into each other, thereby bypassing the neutral switch and allowing the car to start in any gear. Unless you are specifically looking for this it'd be very easy to overlook.
Based on what you've posted already doing with the backup light circuit I'll skip a few things. First, did you check the fuse? It is top left, 14a. You can remove it and check your continuity again, You should have an open now. If not, there's a short to ground in that leg of the harness. Second, am I correct to assume you ran a power jumper to the female pin of the black with red stripe wire on chassis harness that comes out of the firewall (along with the neutral switch wiring)? In addition to what you've done, I'd add turn your ignition to on the take a
power reading on the same connector (chassis side harness), on the male pin black with red stripe wire. If you're getting power there I don't see how it could be anything but a switch problem (e.g. adjustment). One final thing you can do is have someone stand behind the car and with ignition on slowly shift from park through all positions to low. See if the B/U lights illuminate at any time.
Hope that helps give you some ideas!
