Is there a solution for the NES that will allow me to keep the zapper plugged in and usable while other 2 controllers are plugged in?

Solution 1:

According to what I can find online the NES controller port has 7 pins, power, ground and 5 data lines, but only three of those data lines are used by the controller and only two of them are used by the zapper.

http://www.taylorbooth.com/NESController.htm

https://www.beardypig.com/2015/12/11/nes-zapper-experiments-1/

We don't need to switch the ground line, we definately need to switch the two data lines that are used by both zapper and pad and it's probablly a good idea to switch the power line and the data line only used by the pad.

So that is four lines that need to be switched, four pole changeover switches are readilly available and it should be easy enough to cut up a couple of controller extension cables, use a multimeter to identify the wires and then solder them to the switch.