Manifold pressure does not drop when carb heat is applied - it increases. Carb heat reduces engine power, which reduces engine RPM, causing the governor (or pilot) to open the throttle, thus increasing MAP. Manifold pressure increases by about 1-2" when full carb heat is applied. In your example, MAP would drop from 20" to around 18 or 19" when carb heat is removed.
You say that the MP actually increases with carb heat applied and then go on to say that it's the pilot causing this increase. If the pilot is the one CAUSING the increase, then logic points to a decrease from initial MP with the introduction of carb heat.
Reasons I agree that the MP decreases with the introduction of carb heat PRIOR to pilot input? Well what do we know about cold air vs hot air? Cold air is more dense and therefore has a higher pressure. Hot air is thinner and therefore has a lower pressure.
Now I know the first thing everyone is going to say is "nuh uh! When you warm air up it has a higher presure!" When you have a sealed container of air and heat it up, it will in fact exert a higher pressure on the container. This isn't because it actually has this kind of pressure, it's because the air is trying to EXPAND. This expansion is what exerts the pressure on the container.
None of the intake tract is a sealed container. The carb heat is being pulled from outside where there are no constraints on it's ability to expand so it isn't going to magically start building pressure. In fact, the opposite is true since we know that Bernoulli's Principal will affect that hot air as it passes through the carburetor. The velocity of the air will increase and according to said principal, the pressure will decrease even further. "But that's only at the venturi" Also true, but remember, a naturally aspirated engine pulls its air in via a vacuum in the cylinder. The air being pulled in by the cylinder increases its velocity through the intake tract, thereby decreasing it's pressure.
So, with a given MP and the introduction of carb heat PRIOR to pilot intervention, the MP indication should drop. Then it'll rise when the pilot pulls in more collective.
PS I hope that all makes sense... haha. I know it's broken down about as barney style as it gets, but it (hopefully) helped you understand what was going through my brain. It can be hard for me to explain things without breaking it all down sometimes...