I'm trying to wire a 12v bilge manual pump to a

Check the wires and directions, something is crossed. Make sure your pump can do both, be in auto and be used in manual mode. I think the on switch you want to wire in will activate the auto function. There should be a float switch on the pump, be sure that works as it should. The pump may be defective or have a wire crossed.

1. locate where you want the switch (such as inside the RV or on a stake that can be placed near the shower enclosure)
2. If the pump runs on 120 volts make sure the location of the switch is not going to get wet or it may be your last shower ever. If the pump is 120 volts (house current) it may be safer to run an extension cord to the pump and just plug the pump in from the inside of the RV when you take a shower.
3. If the bilge pump is 12 volts just get an on/off toggle and run wires from the toggle switch to the battery and then the red (positive) wire on the pump.

The float switch is normally wired to turn on by itself. To do this, you simply run a "hot" lead from the battery to the float switch, then from the float switch to the bilge pump. It is best if this wire has a fuse between the battery and the pump! The "negative" or ground side should already be supplied if the pump is hooked up to a switch on the dash board. IF there is not switch on dash, you may have to run a ground wire directly to the pump.

Usually the bilge pump would be a 12V pump. So roughly 175W.

Hopefully the float switch for the Bilge pump is under the engine not the pump motor. Having an electric motor under an engine is a fire or explosion hazard Find the float switch for the motor and follow the tubing back to the pump. A replacement pump should have instructions. Only the pump intake should be in an explosive environment.

Does it not have an auto float switch

That sounds more like an aftermarket bilge switch, is the boat all stock?

I was trained at a tracker dealership and that is one of their faults. The best way to find the fuse for the bilge pump is to look on the back of the bilge switch and trace a wire to the fuse panel. Most of the time they are color coordinated so you can match fairly easy without trying to manually probe for the wire.

There's an installation manual available online. Black is negative ground; either brown or brown/white will turn on the pump. The intent is for a three way switch, connecting brown/white directly to positive 12 VDC while brown would go through an automatic switch (e.g., a float switch).