There’s nothing preventing you from setting up the multi stage array and then positioning it with the mirror component active, if you can make sense of that. I simply did it as a two part process because it was easier to understand what was going on.
Easier still is just to create a combined Left/right mirrored spiral segment, and create a single stage array to repeat it radially. That was my original solution before I actually figured out how to do what you asked. Complicated arrays often run afoul of my “keep it as simple as possible” philosophy, so I’m not an expert with them.
It would probably be easiest to bypass arrays entirely, and simply use radial symmetry to mask on the shape with a combined Left/right spiral alpha, then perform a mesh extract to create a radial sequence with thickness as a new subtool. That new subtool could then be positioned to cut into the cylinder with Live Boolean as above. This has the benefit of providing a normalized extrusion from the center of the cylinder, though you’d lose a degree of interactivity.
Good luck!