I believe Randy has addressed this in the past (perhaps the long past - I think in an article in Reefkeeping e-zine). The fundamental chemistry of the solution is stable, since upon dissolution all you have are calcium ions in a high-pH alkaline aqueous solution.
The issue is interaction of the solution with CO2 in the air. That CO2 dissolves in the kalkwasser and reacts with the calcium to produce insoluble calcium carbonate. That's the summary reaction, btw - technically, the CO2 dissolves to form carbonic acid and then to carbonate/bicarbonate ions, and the carbonate reacts with the calcium to form Ca2CO3.
Typically, one sees a thin film of precipitated calcium carbonate on the surface of an unstirred kalkwasser solution after a day or two. To some extent, that film inhibits gas exchange, so less calcium carbonate is formed. I think that Randy tested the conductivity of a kalkwasser solution and concluded that it was still pretty close to saturation over the course of several weeks.
Whether your solution will remain "good" over several months would take some rather complicated technical analysis and some fairly big guesses as to what the CO2 concentration is in the air around your storage vessel. Another fairly big factor is whether you're going to stir up a saturated kalk solution and then pour off the clear solution as it settles into your ATO container. If so, then the calcium hydroxide content will start at saturation, then slowly decline as it reacts with CO2 in the air. OTOH, if the intent is to mix kalk powder directly in the ATO container so that there's some amount of undissolved kalk powder that stays at the bottom, then the solution will obviously stay at saturation for much longer. However, if you do this, be extremely careful that the ATO pump stays well above the settled, undissolved kalk and it can't stir up the undissolved layer when it turns on. A fair number of tank disasters have been caused by pumping kalk slurry directly into the tank.
