in principle, that would work,....any weight moving fore and aft affects the cg. in reality, the system probably wouldn't move a weight enough to make a discernible change in stability. the servo arm, even a long arm would not move the weight enough. moving the weight on a track or plate by closed loop or jack-screw is about the best way, provided the right servo is used and you like to fiddle with apparatus.
possibly a small two-way switch, run by a servo, that controls the rotation direction of a small motor to drive a jack-screw either direction might simplify the idea in that simple motors and small switches are usually allot less expensive than drum servos or 360 servos. it would also facilitate being able to move the weight or what is essentially an unlimited distance, only defined by the length of jack-screw used. end limit switches would, of course be needed.
this system came to me one day while trying to figure out how to trim a sail boat with equipment I had on hand. the jack-screw ran a block on it, that the sail's sheet was tied to.
so in operation, a regular servo controls a directional switch the controls which way the jack-screw motor turns the jack-screw, moving the weight fore and aft .
with this system, the jack-screw could be 6 inches long or six feet long, it didn't matter.

small pulleys or micro chain can then be used to control how fast the weight moves and/or increase the torque of a small, low voltage motor to rotate the jack-screw.