This does seem a bit strange. First of all your main breaker may or may not turn off the alarm
circuit (LED light) when you turn it off - probably does not. It may or may not go off if you turn your battery
switch off IF it was wired directly to the battery or the battery side of the switch. But that means that the bilge
switch should be wired that way too.
I am thinking you are saying that the DC main breaker trips when you turn it on, and not the bilge
pump switch. If it is the bilge pump manual override then it is a different animal. But if it is the main breaker, then yes you seem to have some kind of major short, which could be from almost anything that the panel powers, not necessarily the bilge pump.
If it were me, I would take the time to trace all the wiring
from the panel down to the bilge pump and make a little diagram. Normally the LED would turn on when the bilge pump is activated/powered (and not just that it just has power available - although that is possible). Some people are very anal about knowing that the bilge pump can do its job whether the battery switch is on or not and want to see a light to show that. But normally it would only go on when the pump is activated (whether it pumps or not).
Two ways that happens (if you have a manual override). First is to push/switch the manual override. The light should go on. If you lift
the float switch, it should go on. But I would find the power wire from the battery side (found by your tracing wires diagram). If it comes from the panel somewhere, AFTER the main switch then a short in the wiring to the bilge could be shorted. I would trace the wire to the float switch and see if it is shorted any where.
If it is not the bilge pump circuit you could be seeing a "coincidental" problem, i.e. has nothing to do with the bilge pump.
I've attached how bilge pumps with lights that go on when the bilge pump is energized (or the wires going to the bilge pump whether it is connected or not). Note that the light will go on whether the float switch turns it on or the manual switch is turned on. If you have a dead short in either the wire from the manual side of the switch or the auto side of the switch wiring, then it would cause a trip, but it really should blow the fuse first. You should always have a fuse (put one in if you don't). If you don't then it might trip the main breaker but that is a lot of current
Come back with questions/comments/more description of what you have. Bilge pump circuits are hard to get your head
around if you don't work with electrical