I found a web article, "The Origin & Many Uses of Shellac" by R.J. Wakeman, to be particularly illuminating and well-written source of information on the composition and manufacture of shellac records, for instance it explains:
Emile Berliner’s first disc records were pressed in celluloid (1894) followed by the use of vulcanized rubber (1895).(36) In 1896 Duranoid shellac records were developed and became the standard medium.(11) The formula used for making shellac records varied from firm to firm and from decade to decade; it was usually a company secret. The shellac component represented approximately five-eighths of the material used for the better quality records.(33) Essential components would include shellac, ground rock, carbon black, and cotton floc.
The article lists a few patents among the references that you could quickly search for additional preparation methods, although I found them to be somewhat evasive with regard to details, preferring to amplify the scope of the application by making general claims. The number of patents describing the manufacture of records with shellac and competing resins with a variety of additives is considerable. There is therefore no universal recipe for which to search. What common properties do the formulations share? The key is that the shellac serves as a binder, and that strength and hardness, to protect against wear of the record by the steel needle, to actively wear the needle into shape, or to prevent shattering, is largely conferred by the ground mineral filler and other additives such as abrasives and plasticizers. Apparently the fineness of the ground rock and the quality of the shellac are important.
Abrasives were frequently added to the record matrix to help wear the point of the needle to conform to the shape of the record grooves
and
When a needle enters the groove of a 78 rpm shellac record the needle point should quickly abrade to fit the record groove.
Regarding the preparation of shellac records, Wakeman writes:
To mold shellac records, powdered shellac, ground stone, carbon black, and cotton floc were measured by a formula and mixed in a revolving drum or Banbury mixer.(16) The shellac needed to be finely ground for thorough mixing with and coverage of the other components. High quality orange shellac with low wax content and few impurities was the standard.(16) One report states the shellac was ground to 80 mesh, the fillers to 200 mesh, and the coloring matter to less than 0.4 micron in size. The mixture was steam heated, causing the shellac to melt and form a thick dough which was then passed onto heated rollers and formed into a long roll. The roll was cut into sections (called “biscuits” in some factories) with adequate material to form a 10 or 12-inch record.(8,31)
So in answer to the question, "what alternatives exist to shellac with similar properties?" you would like to replace shellac with other resins, while retaining the other essential additives, such as abrasives. You would not be the first to attempt to do so. As Wakeman explains:
After the war shellac was combined with other resins to form softer and less brittle 78 rpm records.(16) Shellac was reduced in the matrix to form 15 to 30% of the mix.(16) Vulcanite (also called ebonite), a hard rubber, was sometimes included in the mix, as were Vinsol (a pine tree resin) and Manila gum (16).
Another important property is time to harden. I looked up Durium, a relatively simple shellac competitor. Durium was patented as a fast-hardening resorcinol-formaldehyde resin. Production of Durium records was faster since, unlike shellac, Durium hardened quickly and records were fabricated by pressing rather than moulding. While it has properties that represent a significant advantage over shellac (price, hardening speed, method of impression), unfortunately it also requires working with a variety of unpleasant chemicals.
For an early vinyl patent, developed during shellac shortages in WWII, see US patent 2335986. For a more recent patent providing a number of formulations alternative to shellac, see US patent 2787602.
Regarding characterization of shellac composites, this is a difficult topic to address, since the properties can be expected to depend strongly on the specific additives. I did find an old reference quoting a value for the modulus of elasticity of processed shellac as on the order of $\pu{10 \times 10^3 kg/cm2}$. You have many choices regarding a resin to match the required strength and softening temperature. You may want to contact a company such as Ensinger plastics for guidance on selection of a thermoplastic composite.
Again, I'm not talking about Vinyl records, which continue to be made today, I'm talking about shellac.
In general, you would do better if you stop assuming I don't know what I'm talking about.
– Lennart Regebro Sep 25 '19 at 20:48