NOTE*** This is for a 2006 RX400h...I can't guarantee that the wires colors and pin numbers are the same for every model of RX and each year of RX.
Well, I took Carver's advice and found my own solution. All you have to do is cut one wire. Now my windshield washers and headlight washers work independently.
There is a purple wire in the passenger side kick panel that tells the headlight washers when the window washers are on. If you cut the wire, they won't talk to each other anymore. AKA, the headlight washers only work when you press the headlight washer button. If you wash the windshield, the headlight washers don't wash.
In the passenger side kick panel there are four purple wires that ground when you wash the windshield. If you cut the wire coming from pin 10, the washer for the windshield won't communicate with the headlight washer anymore, thus letting your control the headlight washer and windshield washer independently from each other.
There are pictures below of the location of the harness that has the wire you need to cut. Since there are four purple wires in the connector, you must make sure to cut the correct one. If you cut the wrong one, the washer and auto wipers will no longer work.
The harness has two purple wires on the front side (closer to the passenger door) that are pins 6 and 8...and two purple wires on the back side (toward the engine compartment) that are pins 10 and 11. You want to cut the wire coming from pin 10 on the back side of the harness. (This is the purple wire most toward the center of the harness on the backside)
Here are some pictures to help you locate the harness. The most difficult part of this project is the fact that it is in a tight location. My big hands had a hard time fitting in the small space and I have a couple cuts and scratches from reaching up in there.