Unlike the BC-1, which was practically indestructible, the BC-3 has a flexible tube which transmits air from one's mouth to a sensor located in the body of the unit. The tube is attached to the body by slip-fitting over a small plastic nipple protruding from an angle tube that fits into a rubber socket inside the BC-3 body. In addition to having a tendency to slip off the nipple at inopportune moments (usually mid-performance), the tube exerts significant lateral force on the nipple, and this is where the BC-3 typically fails.
Yamaha does not sell parts for the BC-3.

The solution I happened upon is found in 1/4" heat-shrink tubing. This size tubing fits nicely over the flexible tube and the angle tube. Disassembly of the BC-3 is required (two small phillips-head screws old the body together) and the plastic body must be trimmed slightly to allowed the heat-shrink tube-encased joint to clear.
Using two layers of heat-shrink tubing, I was able to reinforce my current BC-3 at the weak point, both preventing the flex tube from slipping off the nipple, and providing additional support to the flex tube - angle tube junction. I have high hopes for lengthy service from this puppy. And I now have a KSM32 metal mic case dedicated to BC-3 transport for gigging, having retired the tupperware to the kitchen.
I also successfully repaired my two older units by gluing the nipple back on the angle tube (with plastic cement), letting the junction set overnight, carefully slipping the flex tube over the nipple, then adding two layers of heat shrink tube over the outside of the assembly. How durable this repair will be is unknown, but my first repaired unit is working fine in the studio.
If you have never tried using breath control, and especially if your keyboard has a breath control jack, give it a whirl. I always get compliments from musicians and audience members alike when I break out the BC-3 brass section.