I replaced an outdoor outlet controlled by an interior switch with a Cooper gfci outlet and it tested fine. Then I replaced the existing switch with a Cooper switch with pilot light.

The switch has one line coming into it with one white(hot), one black(neutral), and one ground. When I connect the white to the brass screw and the black to the silver screw the pilot light is on when the switch is on but the gfci outlet trips and can't be reset. The wiring diagram that came with the switch shows a neutral wire connected from the silver screw to the black screw on the opposite side labeled with a "L" in a circle. When I connect a wire from the silver screw to the black screw the pilot light does not come on but the outlet works fine.

How do I wire the switch so the pilot light is on when the switch is on without tripping the gfci outlet and what does the "L" mean on the diagram?

White is intended to be neutral, but at a switch it can be hot. The black is most likely the switch leg to the outlet. This wiring method is called a "switch loop".

So, you do not have a neutral at the switch, and the pilot light switch you have requires one.

Best to go find a switch with pilot light that does not need a neutral.