That's called relay box 1, there are several relays inside, battery saver relay is one of them
But it reads like your battery is the problem, the clicking means voltage is low in the whole truck, when you open a door that draws more amps from dome light and GEM Module so relay can have even less volts
A good battery has 12.3v to 12.8volts
12.8v new battery
12.5v 3 year old battery
12.3v 5/6 years old and time to shop for battery sales
12.2v or lower is a failing battery, often self draining, works fine when warm during the day, dies when it sits over night
Easy to test with volt meter, battery must have been sitting unused for at least 4 hours to test its "at rest" voltage, so if you charge the battery then let it sit before testing it, it has to HOLD a charge
Also clean battery cables and terminals
Start engine, battery should now show 14.3v to 14.8volts, thats from the alternator working
After idling for 5+ minutes, voltage should be down under 14v, 13.5-13.8volts is expected if battery is GOOD and voltage regulator is GOOD
A bad battery will keep drawing over 14volts