As mentioned in a previous post, 1962 Type 3 rear brakes are different than those on later models. One of the most obvious differences is the presence of an oil slinger on each drum, which uses centrifugal force to direct any gear oil that gets past the axle seal away from the brake shoes and to the outside of the drum. The Type 3 slinger (part number 311 501 631) is a unique one-year-only part only used up to chassis number 0 076 299. Many have been discarded over the years during service, so if you're in need of one, as I am, you're in for a search.
After looking for awhile and following a few leads to dead ends I decided to try another approach. Jason Weigel suggested I try modifying a Type 1 or Type 2 part to fit. Yes, 1950s beetles and buses also had oil slingers, and they're easy to find due to much higher production numbers, so I was able to buy a pair of Type 1 oil slingers advertised on The Samba for a reasonable price. Here is one of them (left) next to a Type 3 slinger from my car (right):
As you can see, the Type 1 slinger is considerably flatter. The Type 3 tube is also slightly longer. I stuffed the inside of the tube with a few pieces of wire to keep it from getting crimped closed and then carefully re-bent it to match the Type 3 part using a vice and a pair of metalworking pliers. The steel was more malleable than I expected.
The idea was to get the tube to correspond as closely as possible to the inside surface of the brake drum, as did the original Type 3 part, allowing clearance for the brake shoes and wheel cylinder. The slightly shorter tube length turned out not to be a problem.
Couldn't hear any rubbing with the drum back in place, so I think it will work. I'll still try to find another real Type 3 slinger but this will get me by for now. Next step: pulling everything apart again to redo the brakes themselves.