That "fluffy thing" is the dampener, also called a "Tone Control."
First Transition Big R Dynasonic Shells numbering D 00001 up to aprox. D 15000 were Oval Badge era shell stock with the new Big R badge mounted in the first panel to the right of the throw off. These drums were manufactured from aprox late 1975 through early 1976, and had dampeners.
Your drum is a Second Transition era Big R snare drum (early-late mid 1976). Your drum has the Big R badge in the new location, two panels to the left of the throw off, and snare drum shells from aprox. 15000 through D 42000 were ordered without dampener prep.
Drums commencing with aprox serial number D 600xxx (later 1976) were drilled for a dampener, and the dampener was part of the COB Dynasonic, with certain exceptions, to the end of production in 1984.
A drum with your serial number should have never had one. There should be no holes drilled in the shell for it.
First.... differences in manufacturing era.
This pic shows a late Big R period, 1980-84 Throw Off, note the rivet top tensioning knob on the throw off. The one you show closeup in the second series of pics is the correct one for your snare drum .
As to the dampener shaft........ This shaft and knob are identical to the shaft and knob on the throw off, and they are interchangeable.
Since a knob on its own is going to be impossible to find, the only source for a replacement is a complete or partial throw off, or a complete or partial dampener. First, remove the throw off from the drum. To disassemble the throw off, you simply unscrew it until the throw off face plate comes off. You will see a very small locking nut just below the U shape of the face plate as you unscrew the knob shaft. Once the face plate is removerd, the knob shaft will still be attached to the throw off body. Carefully grip the locking nut with a pliers or wrench and unscrew the knob shaft from this locking nut, then remove the shaft. Replace and reassemble.
If you are using a knob shaft from a dampener, simply disassemble the dampener, remove the end nut from the knob shaft, swing the arm out of the way, remove the spring, and pull the shaft out of the mounting bracket. Some dampeners used a locking nut on the end of the shaft, others used some sort of method to distort the threads on the shaft end so that the knob shaft would not come apart inside the drum. You may have to file this peened part of the shaft or use a correct size die to recut the threads prior to removing the shaft from the bracket. If you are buying a dampener by itself, this step should already have been done if it was necessary.
This pic shows the inside of the drum at the dampener. The dampener arm and felt were removed by the original owner, The shaft knob and mounting bracket were kept on the drum.
1983 Gina Schock Superten interior dampener shaft and bracket.
1977 COB Dynasonic showing dampener knob
As you can see, the knob on the dampener, and the knob on the throw, are identical.
Hope this helps. I wrote the dating guides for the Big R era Rogers.
http://www.wix.com/theploughman/theploughmansrogers#!resources-and-dating-guides