I did my senor design project on this same problem.
The problem with putting flaps on the main wing is that it changes the airfoil shape of the main wing with respect to the canard. Deploying flaps causes a change in camber of the airfoil. This changes the stalling characteristics of the airfoil. In a canard aircraft it is important that the canard stall before the main wing and that the main wing go through zero lift before the canard. When flaps are deployed on the main wing, the main wing now stalls before the canard and the canard will go through zero lift before the main wing.
You are also correct in that the flaps also add an additional pitching moment which tends to over-load the already heavily loaded canard. The starship had the moving canard but if you listen to the pilots that flew it the flaps really didn't do anything. The problem with moving the canard forward is that it also changes the neutral point of the aircraft. Thus if the canard moves forward as the flaps are deployed the cg also has to move forward or your static margin goes down, thus it is like a dog chasing its tail. Plus you still have the stalling characteristics to deal with.
The way to get around it is to put flaps on both the main wing and the canard and rotate the whole canard for pitch control. The flaps on the canard have to be deployed at a high rate than the main wing's flaps. This solves both the stalling issues and also the added force placed on the canard.

Nate