Shimano Inter-3 Hub coaster brake maintainance
One I dealt with, I think a 3C40, that came in with a complaint of squealing while braking turned out to have had broken chunks coming off the part known in the Shimano exploded drawings as the "Driver Unit", which was replaceable. Based on that I recommend just opening it up, ideally after getting some of the Shimano internal hub grease. It's a little fiddly but not very technical, and Shimano's instructions are good. If you really want to try to address it with minimal disassembly, you could probably get some grease to reach the pads by taking off the left side locknut and cone/reaction arm. The pads reside just a little further in on the left side.