Switching a 12V AC current from a microcontroller (no magnetic relay)
Any SSR will have a significant forward (on) voltage drop of 2-3 V. 
Typically where two triacs (one opto-isolated and one power) are used there is a very significant turn on voltage required. While I have not seen one needing the 75 V specified for the Q3MB-202 I have seen plenty that don't turn on till about 20-25 V.

I'd suggest that you could use the MOC301x or MOC302x optically isolated triacs.
These will turn on with only 3-4 V but at the current levels you want they will typically drop about 1.5 - 2 V RMS. If you actually want to get 12 V RMS on your load you may have to use a 15 V RMS transformer output. 

You can parallel 2 * MOC30xx triacs and they will work well (you could expect less on loss). BUT ...you must have both the LEDs on all the time you want the AC on.
You can parallel the outputs and you can either operate the LEDs in series or drive separately with their own limiting resistors.

Another possibility is to use a DC SSR. These typically have a FET output and you could use a low Vf Schottky diode bridge to switch the AC. This would have a slightly lower loss voltage probably around 1.5 V, though there is not a lot of gain.