As you've bought a 'centre off' switch, you can use that on its own if you like. No need for a separate on/off switch.
You will only need to use one side of it; an SPDT switch was all that's required as a selector here.
Wiring it up is simple enough; centre terminal to the lights, and the upper and lower terminals to the AC and DC+ supplies.
Just keep in mind how the switch contacts work, as to which supply is 'on'. I.e. toggle up = lower terminal supply is on, and vice versa.
An issue I'm unsure about here, is the suitability of the LED light modules [if that's what you've got] for direct AC feed.
The AC supply will need to be half [series diode] or full wave [diode bridge] rectified, if they aren't.
