Before you condemn the gearing inside the box, be sure that the drive pin in the furling drum is fully engaging in the drive plate that is mounted on top of the gearbox. We had our drive plate blow apart a few years ago, and other people have had trouble with the hole rounding over and pushing the pin up. That sounds like the more likely cause.
Having just had our furling gearbox apart for routine maintenance, I think it very unlikely that the gearing inside the gearbox is the problem. Certainly possible, but unlikely given what you describe as symptoms.
For the manual furling line, we have 10mm double braid dacron. That worked great for us on the two occasions we needed it. I wonder if the StaySet might be too stiff? 7/16" would not be too big.
If you need this unit disassembled and looked at, you might try Nance and Underwood, riggers in Fort Lauderdale. They know Amel's very well, and Roger Underwood has been a help when we have used him. If you need a new drive plate, I know they can make one for you.
Fort Lauderdale, FL, USA