That should work fine. With respect to how the Dual Ramp Timer works, none of us have the internal circuitry so we can only guess at how it really functions. If you look at page 1 of the instructions, the diagram shows "12-24VDC Transformer" for both hook up configurations. There are no warnings or cautions regarding using one voltage over the other. There are plenty of DC to DC voltage converter circuits that can reduce a given voltage without the use of large heat sinks or heavy transformer coils, particularly for low power applications such as this. Could be as simple as a zener voltage regulator diode, or more complex. In any case, getting the 12 volt supply is your best bet. You can use it on your LEDs without the Dual Ramp Timer if you wish.
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