We study the radio properties of the optically luminous ``AI'' subclass of quasars at MV
=(1)/(2); Teerikorpi 1981A&A....98..309T
; Papers I, II), after first results on double sources in Paper I and the confirmation in Paper II that optical activity abruptly increases when one crosses the gap around MV
=-25.5 between AI and the optically fainter quasars. AI double sources define the upper envelope in the size vs. redshift diagram, so AI quasars are not only optically the most luminous, but also in radio the largest objects in their redshift range (from 0.5 up to at least 1.6). They are typically good FRII objects, mostly with a small lobe-distance asymmetry Q. A part of the scatter in size is due to orientation, which reveals itself via the size vs. core strength dependence, such as is expected from relativistic core boosting with Γ≃5-8. This relation appears for symmetric sources (Q<1.5) and especially clearly for sources with lobe spectral indices either shallow or steep (``Kardashev''). AI sources occupy the upper envelope of the latter core-size relation. Asymmetric sources (Q≥1.5) or those with intermediate lobe spectral indices 0.7≤αL
≤1.0 do not show any core-size dependence. Their bending angle does not converge towards a small value for large sizes, as it does for the sources with a strong core-size relation. The data suggest (as in Paper I) that the asymmetry Q increases from AI to the edge of the fainter population, which, down to MV
≃-24.8, contains unusual double sources, with a more versatile and less symmetric lobe structure, often with a hotspot in the trailing edge.