Does the operation of a cylindrical microwave cavity resonator as described here, https://en.wikipedia.org/wiki/Microwave_cavity, depend on the speed of light being isotropic along the cylinder axis?
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No. Switching to a coordinate system in which the speed of light is not isotropic has no effect on anything except the complexity of the maths.
Probably relevant here is that Maxwell's equations are different expressed in the two coordinate systems. ##\partial/\partial x## is messy when transformed into coordinates where space and time are not orthogonal. The result of the mess will be more mess, but any prediction you make about instrument readings will come out the same as if you had just done it the easy (isotropic speed) way.
It depends very much on the functional form of the anisotropic light speed assumed. One particular anisotropy (the usual one assumed in these circles) is equivalent to a coordinate transformation and thus is not a physical (or observable) effect (it's not even an effect actually). However, not all anisotropies one might assume are removable by coordinate transform. For these the underlying physics is changes and so is pure speculation.