@David: you are right that the AC controller will work fine with 9VDC (and since it has internal diode protection it won’t be damaged even if the polarity is reversed). In fact, in the past we’ve been customizing the AC controller for a few users who needed a DC version, but basically replacing the triacs with NPN transistors (or MOSFETs).
The MOSFETs are actually typically quite small, because MOSFETs have very low on resistance so as long as it’s fully on the power consumption is usually very low, even if it’s running a large amount of current.