Configuring BLDC motor for low-speed application
Side loading is probably an issue with a motor this small, but here is the real problem: if the rod "weighs" .96 N, and the center of gravity is .102/2 or .051 meters from the shaft, you have a torque load of .049 N-m on the way up.  Even if you find a motor large enough to provide this torque,  it won't run at a constant speed because you will have this torque opposing you on the way up and aiding you on the way down, so you won't get a sinusoid with any small bare motor without feedback.  You could go with a stepper or possibly a brushless or brush motor with a gearbox and hall feedback, or if sensorless, very good speed control. By the time you get to a motor or gearbox capable of providing the torque, you will likely have bearings that can handle some side loading.