Switching Between Two Circuit Branches Using Basic Components
Assuming that your existing circuit works well enough as a single-output, I'd suggest adding a PNP transistor stage to follow on and add a second inverted output.







simulate this circuit &ndash; Schematic created using CircuitLab

For your original circuit, when its light and the LDR has a low resistance, the voltage divider it forms with R2 will turn Q1 on which pulls OUT1 low. When its dark enough and the LDR resistance rises enough to cause Q1 to turn off, OUT1 is then pulled high by R1.
The follow-on stage with Q2 acts like a simple inverter using OUT1 as its input. When Q1 is on (and OUT1 is low) Q2 is turned on by the current flowing out of its base through R3. When Q2 is on, OUT2 is pulled high and when Q2 is off OUT2 is pulled low by R4.
So although both transistors turn on and off together, the way they're configured causes OUT1 and OUT2 to be in opposite states - much like the SPDT switch you're thinking of.