From the photos and video, the stepper unit plastic gear looks dirty. Make sure there is no lubrication or grease on the teeth of the plastic gear or the tips of the gear levers. On the other side, make sure all the contact rivets are clean of dirt and any lubrication/grease.

If it still doesn't work after the cleaning, and you've done the recommended cleaning as mentioned at the PinRepair Web site, then you should look at worn mechanical parts, such as the gear levers, coil plungers and coil stops, etc. This could be a reason why the problem mostly happens during the double bonus mode. The quick double pulses of the solenoid might be causing a slipping effect...

It's rare a spring would go bad on one of these steppers. I would try to troubleshoot the problem with the original springs by first looking for dirt issues and wear issues...