One method I've used in the past to make a season-synchronised 'tick' (which could be vital aspect to an automated melting/freezing machine, so you aren't sending magma under unfrozen water, or whatever mathoed you were planning) is unfortunately a water-user itself. (Not necessarily to loss, but needs some 'feed', and there could be evaporation.) But perhaps you can build it into your water-multiplier anyway in some way, with the following brief description.
Water from an internal location (e.g. from out the side of a hill) is let flow into the outside (or into a specially opened pit, if you don't have any decent hillsides), and flows down, probably in your case to be re-channelled back underground. When external temperatures plummet to below freezing, this forms a plug, which backs up the outflow and activates a (still inside, but 1Z higher, on the level where the water actually came from, e.g. top of a pump-stack) depth-trigger, indicating freezing conditions. When the plug unfreezes, the water level drops again, indicating non-freezing conditions.
You can probably work out the specifics of how this would integrate. I suggest putting the trigger in a normally dry alcove (on the pump-stack level) off to one side of a channel (into Z-1 from the pump-head) that normally takes all the available water. The plug stops this draining and the pump fills the channel up to the second (pump-height) Z level up, as well as the alcove.
Assuming any evaporation in that cycle isn't depleting more than your ice-works is generating, it should be Ok. Otherwise a thermocaust below unfrozen water could just as effectively be cycled up and down all the time, only of course without any water-gain at all, for the times the water stays liquid. So maybe you don't need the thermostatic trigger, anyway.
(Several further replies, while writing, posting anyway, reading these momentarily...)