Control motor braking with PWM
When the brushless DC motors are turned by the force of the fibers, they act as permanent-magnet synchronous AC generators. If they don't need to turn on their own, they can be connected to a rectifier loaded by a resistor with the DC load current modulated by PWM. If the PWM is controlled with feedback from the load cells, that should work.
If the motors need to be turning on their own without the pull of the filaments, the problem is more complex.