soldani..
I can't really tell from your descriptions what's going on...it's not easy to describe situations in unambiguous terms. As noted, the lamp should not affect pump operation; it is wired in parallel with the pump. Seems like you have the right idea: if the pump runs in one position and not another, then it's the latter circuit where the fault resides. Perhaps its a broken wire or corroded connection, not a fuse issue but who knows how aftermarket stuff gets installed .
If someone holds the momentary "on" are you SURE the pump is actually running??..can you hear it spinning or see water discharge??? Sometimes a toothpick can hold the switch "on" while you inspect below.
If the float does NOT turn the pump on, then there is a problem somewhere between the float switch to the pump or float switch to the flybridge switch. Don't assume the red pilot light means much unless you know where it's connected.
I don't personally like the rather popular on/off/auto bilge pump switches...too complicated and too many wires. But I guess if the float switch fails it does provide a way to turn on a pump. The few such switches I have had I never used except in "auto" mode. Some of my bilge pumps don't even have on/off switches...just a fuse,switch pump....unambiguous wires with everything in series. I don't want any bilge pumps accidentally shut off.
An advantage of a switch is you can turn the pump on and listen. I prefer to crawl to each float switch, lift and see if the pump actually comes on..once in the spring and again in the fall...that's the only failsafe method to assure each pump works properly. It also enables me to check there is nothing blocking the pump intake.
Anyway, the indicator lamp SHOULD be wired to the positive of the pump right at the pump...the closer to the pump the better.....so it should ideally mean "lamp on: pump has voltage"...but if the lamp connection, say on the flybridge, is actually connected to the output of the switch to the pump 30 feet away from the pump it's not such a good indicator. I don't know how Hatteras does that. But the lamp can't tell you the pump is running...you have to listen/look at the pump or observe water discharge for positive verification.