GL1800 reverse engages for a fraction of a second and quits, reverse light then goes out.

When you look under the mower do you see two belts ? Top one goes to the rear axle and lower one goes to mower deck spindle.
Your description sounds like the electric clutch is engaging,so if you are seeing two belts coming off the bottom of the engine maybe the belt bounced off a idler pulley on the deck and is not tight enough.

Turn the key on and see if the light comes on, it should. If it doesn't , then someone has taken the light out or the ECM is bad. Put it on a scanner and have the codes read.

impossible to actually see if lever is Fully engaged---with door off try to engage with a match stick or something not metal--see if set comes on that way.

reverse relay needs to be replaced.

that box does engage the transfercase to 4x4,there is a small electric motor on the transfercase thats probably bad,try hitting it lightly with a hammer because just like an electric window motor it has brushes in it as well,but its possible that motor is shorting out causing it to draw alot from the charging system,at least my 1993 Ford Bronco did the same thing and thats what i replaced and now it works fine.