If the plane was told how much weight was onboard, and it knows what throttle it was at... then it should know what acceleration it was expecting.
If it accelerated slower than expected, then there must be something wrong! So abort the takeoff...
Planes should have alerts for this kind of thing. If, 1 second after throttle is applied, the plane hasn't achieved the predicted speed, then the takeoff should be aborted - by this stage, you'll be well under the maximum abort speed, so you can stay safely on the ground and figure out whats wrong. Being overweight is just one possible cause - it might instead be that an engine isn't producing sufficient thrust, and that would probably doom the plane.