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And, of course, after asking the question, I found the answer.
For the benefit of future me, as well as others looking to do the same thing, you can, indeed, retrieve this for the active extruder stack. For example:
# Get the active extruder stack
extruder = ExtruderManager.getInstance().getActiveExtruderStack()
# Get the material for the active extruder stack
material = extruder.material
# You can retrieve the brand, material, and brand from the material metadata
metadata = material.getMetaData()
brand = metadata.get('brand', '')
material = metadata.get('material', '')
color = metadata.get('color_name', '')