What could be causing fuse to burn out when I shift car into reverse?

Try installing a pair of spade terminals in place of that fuse, then use a pair of small jumper wires to connect them to a 12 volt light bulb. Now, when the circuit is powered up and the short is present, the bulb will be full brightness. You can unplug things and move wire harnesses around to see what makes the short go away. When it does, the bulb might go out completely, but more likely it will be on dim because other circuits will be drawing current.

Unplug the neutral safety / transmission range switch, and check the wiring to the backup lights to see if any are bare and touching.

Possible bad or shorted rev. Light bulbs or sockets. It goes from the neu. Switch to the rear.

All you have to do is whatever it normally takes to cause the fuse to blow. If that happens without running the engine, you won't have to run the engine.

If the short is gone when you unplug the neutral safety switch, all you know is it is somewhere between that point and the backup lights. I never autopsied a switch to see if it shorting is possible, but regardless, a pinched, bared, or rubbed-through wire is more likely to be the cause. You can forget about the other things in that circuit because if one of them was shorted, the fuse would blow all the time, not just when shifting into reverse.

The closest book I can find is for a '92 Buick Park Ave, and the transmission range switch is different, but if they use standardized wire colors on all models and years as most manufacturers do, the pink / black wire brings 12 volts down to the switch, and the light green wire sends that current to the backup lights. There could be a light-monitoring module in that part of the circuit too. The wire colors could easily be different on the switch, but if you have those color wires in the electrical connectors to that switch, you can unplug those connectors, then use a jumper wire or stretched-out paper clip to connect them. That will totally remove the range switch from the circuit. If the test light is still bright, you'll at least know the switch is not at fault. If the test light is dim and the backup lights come on very dimly, you'll know the circuit is working. At that point you can replace the test bulb with a fuse and the backup lights will be full brightness.