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Having dug into the source of the arrangements package, I see that the constructor for arrangements sorts the supplied list of hyperplanes, which is what causes the scrambling mentioned in the first bullet point. I believe this will happen both when I construct the arrangement I start with and when I restrict, so I decided it was not worth the trouble of unscrambling these sorting operations.

Instead, I tried to use the workaround of the second bullet point, just using the equations of the hyperplanes (which I know) and evaluating them at representative points of all the regions (which I can delegate to Sage). At this point, it was simpler just to directly work out equations for the hyperplanes in the restriction and construct the actual hyperplane arrangement object from that. For reference, though, the way Sage does this automatically is to choose the variable with the largest coefficient in the equation of the hyperplane H to restrict to (breaking ties by choosing the first such variable) and eliminate that variable from the equations of the other hyperplanes by subtracting off a multiple of the equation of H.