I had a Black Dog and AII Stormy Monday combo, not in an SG but a lightweight Flying V, so a pretty similar all-mahogany tone.
I wouldn't say the BD was "extra dark" in that context, in fact on its own it sounded good, but I didn't think the combination of the dark-ish BD bridge with the relatively "light", clear SM neck worked particularly well. There wasn't enough contrast between the two.
From your initial description I think you'll like the SM neck (maybe Mule if you want a bit more balls, but since you like the SH-2 neck the SM should suit you fine). But maybe something a bit brighter than the BD for the bridge.