Sorry for the delay, but about three weeks ago I turned on the pump in the sauna on a lark, and lo and behold! Water!
No more carrying buckets from the cabin to the sauna! That's awesome! I am planning to reroute the plumbing in the sauna and remove the storage tanks there eventually, and simply have the water pump into the sink whenever we turn on the switch. Now that we have a decent supply in the cabin, we really don't need to store much water in the sauna.