I double-checked my info, it does seem that the range recognition switch would be the most likely cause. It sends a voltage-coded signal to the ETC (transmission control module) which in turn supplies the voltage for the reverse lights. Since it's voltage-coded, it would be especially important that the contact for the switch is clean; you haven't spilled anything into the shifter, have ya?
I'd break down the difficulty into removing the shifter from the car as maybe a 3, and replacing the range switch as around a 7 to 9 or so, depending on your patience. Drops considerably if you can get someone to get the instructions to you from MB (WIS)