What he didn't accept is that the speed of light is the same for observers in all frames, and that is not an empirical fact.
He did more than "not accept" the empirical fact that the locally measured speed of light in vacuum is always the same -- he asserted that the speed of light measured from a source moving at speed v relative to the observer was c + v (classically ballistic light particles), which has been found to be specifically untrue in the case of stars moving at many km/s and pions moving at appreciable fractions of the speed of light as well as being unsupported in other areas of electromagnetic phenomena. He would cherry pick sources from the fringe in support of this idea to the exclusion of all other data. Science doesn't require him to accept special relativity as some sort of axiom or tenet of belief, but methodological empiricism requires that Wallace accept that special relativity is an extremely precise summary of all relevant empirical fact when he has no facts demonstrated to the contrary just as empiricism requires him to reject the Newtonian theory of ballistic light which he clings to despite the evidence and despite that his objections to quantum electrodynamics are grounded in an aesthetic rejection of some pop-science straw-man description of the physical theory.
Using the test theory intermediate between c and c+v, many experiments since 1913 have been done to try and directly find evidence for what the value of k is in c + kv.
K. Brecher, “Is the Speed of Light Independent of the Velocity of the Source?”, Phys. Rev. Lett. 39 1051–1054, 1236(E) (1977).
Uses observations of binary pulsars to put a limit on the source-velocity dependence of the speed of light. $$k \; < \; 2 \times 10^{\tiny -9}$$
So Wallace is wrong on the order of a billion sigmas