I think the way to wire this would be to use a double pole, double throw switch (DPDT).
Think of wiring the switch as follows (as in how to hook up to the contacts in the switch):
Position 1 (from source)
Source (-) Source (+)
Position Middle (to light)
Light (W) Light (B)
Position 2 (from source)
Source (+) Source (-)
An easy way to do this would wire source to position 1 and then jumper position 1 to position 2 but crossing the wires (position 1 (-) to position 2 (+))
Depending on the way the switch is thrown (position 1 or 2) determines the polarity of the wires leading to the switches. Both the wires are hooked to both lights, the bulbs installed with opposite polarities so that only the LED light fed with the correct polarity will light.
The switch could also have a middle position where no contact is made.
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