I'm doing what you are now investigating doing, and have had great success with it for 15 years, except that I only use one dosing pump from the Kalkwasser reactor to the sump. Water level is maintained in the Kalkwasser reactor using a Toilet float valve.
It is neither 100% air sealed nor do I use a stirrer (Randy Holmes-Farley, if I am not mistaken, has commented why the use of non-air-sealed isn't such a big deal). I use an 2600lph Eheim pump, and I use some stiff curved vinyl tubing to direct the flow to the bottom of the salt bucket I use, so as not to disturb the crust on the surface of the water too much.
I drop in a kilogram of dry kalk every 3 months, and the pH in the reactor stays at or above pH 12 for about 2 months. After 2 months, it'll start to drop below to somewhere between 10-11, but that last month, I increase the time to mix the sludge at the bottom, from 2 mins to 4-5 minutes which pumps it back up to 12. By the end of month 3, I have to recharge with another kilo of Kalk.
What I have in my setup that I do not see in yours is:
1.) I use a lift bottle (recycled cola bottle & bottom cut off) in the Kalkwasser reactor with about 1-2 inches of the top of the bottle above the water line. This is in case my pump were to fire at the same time (accidents happen) as the dosing pump were running, it'll keep most of the sludge / undissolved kalk from getting sucked into the dosing tube
2.) A sediment bottle - a sealed glass cola bottle inbetween the reactor and the dosing pump as a final catch all of any floating sediments from the Kalk. Over about a year of dosing, it always has 2-3 inches of sediment in the bottom.
Once a year, I tear it all down and clean everything out. The buildup on my Eheim pump isn't really an issue at all, and usually my attention is to my float valve. Everything gets a good wash and scrubbing with vinegar water, slapped back together and we're good for another year.
I have my controller setup to spread out the approx 6 liters of dosing Kalk as TopUp every hour during the night for 12 hours, to help reduce pH drops at night. During the day, it switches over to RO for the rest of AutoTopUp.
FWIW, my total system volume is about twice as yours, so I can't totally rely on Kalkwasser, so I run a Baling Lite / Kalkwaser Hybrid setup. Kalkwasser takes care of the majority of my CA / Alk needs, but I fill the small gap left over with minor dosing of CA & ALK via Balling, plus my Mag demands.
