So I'm using a timer in my game to track the players best completion speed, and the way I do it is using an autoload with an incremental counter code that's basically the following:
var frame_count = 0
var active = false
toggle(onoff):
active = onoff
reset():
toggle(false)
frame_count = 0
_physics_process(delta):
if active:
frame_count += 1
If you're using the built in pause function, you can just set the autoload's pause_mode in ready. If you're using a custom pause function, you can just use timer.toggle(x) from anywhere in your game. I know it's not using a stopwatch, but this works well for me.