I wonder how big CaCO3 particles are we talking about, ie what are the approximate amount of coating vs particle. It significantly influence what is the right tool.
Since the Ca salts of acids are rather stable and less soluble compared to Na or K salts, I see why such a system would be stable. But cannot just wash it down / rinse eg with a concentrated solution of some soluble Ca salt or other cation? Mechanical stiring, ultrasound etc may be very important if you go this way.
Also, are these separation methods the best way to approach the problem? Your acid is a really dirty material in water, hard to make nice reactions with it, produce foamish surfaces, may stick on your glass etc. So in short I wouldn't bother with taking it down and trying to measure it. Have you though about something simple like gravimetery / TG?