Jim,
At what point were you wanting to attach the sling swivel?  I would think You would want to be on the actual wood and not the black extension. With the thin area under the magazine box I think the attachment point from the barrel and stock ought to be close together if not 2 widely separated points of stress could bend or break the stock in the weak area.  You never plan to fall when carrying a rifle but it does happen.
Parashooter's idea with the smaller Win #67 dovetailed nut is not a bad solution. Out there around 8 or 9 inches the barrel is even thicker and if it is also OK for your swivel location it could work.
Another thing that was done in days gone by was installing a "beauty ring" next to the receiver, it was a steel tapered tube and some had lugs welded to the bottom for attachment a point for a screw was run through the bottom of the stock. the beauty ring made the transition from the receiver diameter to the barrel a little more graceful but it would be a long way from your swivel attachment point. 
Will you attempt to do this your self or have a gunsmith do it? Do you already have files, drill bits, a hand drill, maybe an old fashioned brace and bit, a vice big enough to hold the stock plus padding. There are a lot of ways to skin a cat, I think I could do as Parashooter mentioned, file a flat bottomed groove in the barrel then do the dovetail cuts with a triangular file with a ground safe edge. If you have a nearby machine shop they will have bar ends you could practice on if not I can send you a couple. If you do not already have all this stuff, it might be less expensive to have a gunsmith do it.