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The LW special is itself an enchant, not a socket. That's like saying you can't get the inscription shoulder enchant to stack with the current one.
I don't know for certain, because I haven't seen the tooltip for the socket. But every profession has what I term a "special" that is for the user only. For LWs it's Fur linings, for engineers it's the glove and boot add-ons, etc. The LW one isn't an enchant, it's a "fur lining".
From my experience, normal enchanter enchants do not stack with the profession specials. I do not have experience with the BS socket specials though. I'm gonna guess they don't stack, but the only way to know for certain is to try it. I'm sure a tooltip will ask you to confirm if it would actually replace teh other rather than stack.