First of all this word "inertia" is bandied about too loosely causing much of the confusion. It has at least two different meanings in relativity.

Meaning # 1  Origin of inertia in one of Sciama's senses is trying to explain why test particles move on geodesics.
i.e. zero g-force free-float "weightless" "inertial motion" universal independent of the rest masses m of the test particles.

This is essentially trying to explain Newton's first law of mechanical motion generalized to curved space-time. Einstein with Infeld actually proved the geodesic rule from his nonlinear field equations

Ruv + 8piGTuv/c^4 = 0

modeling the test particle as a singularity in the 4th rank curvature tensor Ruvwl field if I remember correctly off the top of my head.

The Einstein-Infeld calculation is purely local not needing Mach's Principle.

Mach's Principle is archaic and quaint referring to "distant matter" that makes no real sense at all in modern precision cosmology since the 1990's because the only kind of "matter" Mach knew about is only 4% of all the gravitating stuff in the universe!

Woodward refers to the distant matter in "causal contact" with his local flux capacitor space drive prototype device that he actually has running in his lab much to his engineering credit. On the other hand, the situation seems to be like cold fusion and high-frequency-gravity-wave propulsion & Podkletnov type allegations discussed at the JASON meeting I attended a few years ago at General Atomics in La Jolla - hard to replicate by independent parties.

Woodward is not clear what he means by "causal contact" at times he seems to mean only the past light cone out to the "particle horizon" of the flux capacitor, but this contradicts the key assumption in his eq (44) that the universe as a constant density of "distant matter" if by that he means what Mach meant the "distant stars" - there is great ambiguity here and his equations rest on very shaky ground.

Keep in mind Tamara Davis's diagram

At other times Woodward alludes to Wheeler-Feynman and of course Sciama worked with Hoyle who developed the Wheeler-Feynman classical retrocausal ideas to quantum theory and cosmology. This brings in the future light cone of his device reaching to our future event horizon - hence John Cramer's "transactions" and Yakir Aharonov's "destiny" (post-selection final boundary condition) and even 't Hooft-Susskind's hologram conjecture.

Meaning #2 of "inertia" as resistance to non-gravity forces pushing the test particle of rest mass m off its timelike geodesic determined locally by source stress-energy density tensors Tuv(non-gravity fields). No need for Mach's Principle here.
So this is essentially F = ma Newton's 2nd law of mechanical motion of test particles generalized to curved spacetime.

The computation of the rest masses m of ordinary matter (real on mass shell in sense of quantum field theory's Feynman propagators) does not require gravity or Mach's principle, but is explained quantitatively for hadrons by Franck Wilczek's quantum chromodynamic supercomputer computations using the Higgs field couplings to the quarks in the input to the program.

What Sciama means by the "origin of inertia" is really the origin of the universal geodesic inertial motion of neutral test particles. He does not mean that Mach's principle is needed to explain why or how the rest mass of the electron is ~ 10^-27 grams or the rest mass of the proton is ~ 10^-24 grams - that is not a gravity physics problem needing the cosmological scale Mach's Principle.