Tramp,
I think you will find most of the China coaster brake hubs are identical just different names on the brake arm. So most any thrift store or yard sale hub is your spare parts.
This certainly does seem to be the case; the KT hubs (for example) are of an identical desgn to the Shimano CB-E-110, with a few exception on the brake side, but the internal dimensions of the hub are the same. I haven't tried on a Falcon-branded hub, but my inclination (if you want new parts) would be to buy a new complete cb-e110 hub, and see if the guts don't swap. It IS a gamble, b/c if the hub shells' ID don't match perfectly, or if the way the shell was milled to interface with the clutch cone is off by a mm or a degree, then it won't work. But, if that's the case, you can just lace the new Shimano hub into the existing wheel.
BITD, they used to sell a repair kit for the CB-E110, which was a complete hub, minus the shell and the trim kit. (Typically, even complete new hubs don't include the trim kit, which includes the cog, snapring, dustcover, and brake strap--something to keep in mind while ordering...)
The thing is, the repair kit was only like $2 less than a complete hub, and in a situation like yours, you might be best off having the hub shell, just in case your Falcon isn't compatible with Shimano guts.
The probable reason why they don't offer repair kits for the Falcon hubs is b/c these are oem equipment on low-end bikes. I'm not certain that you could even buy a Falcon coaster hub as a stand-alone aftermarket component; they are pretty much the original equipment for low-end dept store bikes, and are generally considered to be a consumable/disposable part by virtually everyone outside of the RRB community. The Shimano hub is no prize, but you can buy virtually every part for them a la carte, and they are noticeably superior to the KT HiStop, Falcon, CSC, etc...On the occasions where i swapped the KT internals for Shimano bits, you could feel a difference in weight and harness on many of the Shimano bits; they were also knurled differently where it counts. So, in the battle to decide which of the turds is greatest, the Shimano cb-e110 is the king of the toilet.