No, still reservoirs should never be stirred. That will add CO2 to the solution and cause even more precipitation. Saturated kalk can remain (mostly) potent for weeks if you keep a lid on the container.
Kalk reactors are different, the RO/DI water is fed in and mixed occasionally before getting dosed into the tank. A still reservoir is already mixed.
Hope that helps.
PS: Personally I just add pre-stirred mixed kalk into the ATO reservoir. Any extra kalk settles at the bottom, the saturated limewater goes into the tank.