Use arc3D[[1]] (which contains all the graphics primitives and their styles) as the first argument of GeometricTransformation:
arc3D = ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t, 0, π/2}]
Show[arc3D,
Graphics3D[GeometricTransformation[arc3D[[1]], shiftAndDilate3D] /. l_Line :> {Orange, l}],
PlotRange -> All]
