I would screw an L bracket to the speedring insert, and attach that to the swivel or ball head etc. That would be a pretty easy DIY for anyone with a hand drill (and a shelf bracket etc).
Its a really nice looking softbox. Though all the plastic in the speedring mechanism etc still looks like it could be a bit of an issue down the track even without the speedlight bracket section needed.
Ironically Elv, using an L bracket as you suggest would put the 27 Oz CL-360 weight stress on the mount in the head rather than the 13 Oz weight of the 70cm modifier.
I think Dave's approach makes good sense and won't stress the mount. One should remember that the tube part of the mount slides into the head about an inch or more. That's a lot of surface to distribute the weight. The large pad on the 1/4 20 fitting on the ball head, or the padded spigot shown offer good distribution for the weight of the assembly on the 1/4 20 fitting on the head.