What happens if my SSR output voltage is rated higher than than my expected throughput?
I would just try it. Maybe you're lucky and it will work for you.

If the manufacturer could guarantee that it would work with 12 V AC on the switched side, I have no doubt that they would put that in the datasheet and print it on the component. But they did not.

So there is no guarantee that it will work. Since the voltage you want to switch is below the rated voltage I see no harm in trying.

The reason why it might not work is that there is a switching element inside, probably a thyristor or a TRIAC. To switch these on properly (with a sharp off-to-on transition) some voltage is needed.

Thyristors and TRIACs only switch off when the current through them becomes zero so that is why this device will only work on AC. If you would switch a DC current, the switch would be on until you remove the current.