I don't use a sling pack, but I use a hip pack and had a similar issue with carrying a net. The solution that I ultimately came up with and am happy with is a homemade "net holster" of sorts.

I picked up a plastic conduit coupler at my local hardware story. It's basically a plastic cylinder about 2" long and large enough in diameter to fit the handle of my net. I fastened the coupler to the belt of my hip pack in a vertical position with a couple of zip / cable ties. My net rides handle down in the holster with the bag up. It's quick and easy to pull the net out when I needed it and to replace it when I'm done with it. I haven't had a problem with the net falling out accidentally, but I tied a piece of shockcord to the net and then secure that to my pack just for safety sake.

I think you could apply the same idea to a sling pack and your net, but you probably need to experiment and customize a bit.