I hadn’t thought of using regular loudspeakers, but you may want to look into voice coil actuators. These are precise enough that they are used for hard drive RW heads.
I’m not sure what shafts you mean, but maybe you can use dowel pins. These can be had with diameter tolerances of 10um or less, as standard parts.
The rubber tube I used was a slingshot replacement, so it was designed to be stiff. In practice, you would use silicone rather than latex rubber (more stable), you would specify thin not thick wall tubing, and there could be a compensating spring to enable cheap (not powerful) stepper motors to be used to actuate it.
After the main payload has been dispensed, the last bit could be
spat out of the pipette tip by squeezing or rolling the now-flaccid tube, like dispensing toothpaste.
Looking forward to your results!