Deffiantly KEEP the tree... I love dead trees as much as I love live ones!! If this was my run I would buy welded pig fence panels (I'm actually gonna use them for my pernament run already, I just won't have a cool tree sticking out the top). They come with smaller openings like 2 inch so it would be small enough gaps to keep the fox and dogs and what not out. The panels are sold flat, not rolled up, so they are more sturdy/stiff then what you can buy in a roll. Attach both ends of the fencing to either side of the top of your run walls (maybe on the side that doesn't have 2x4 running horizontally you could use zip ties or pieces of wire to secure it to the exsisting fencing so you won't have additional costs, I'd just use a staple gun across the side where the 2x4 is) so they create a dome shape over the run. If you space the fencing just right and cut what space you need out of the sides (would be the top/bottom of the panel if you think of how it would be used traditionaly to build a fence) of the fencing panels to accomadate the tree then you'll be able to staple it directly to the tree to secure it. Hopefully you can find tall enough panels that one panel will go from the end of your run up to about half of the width of the tree. That way you will be able to cut the space needed to accomadate the tree on the side of the panel so you won't have the need to go over the top of it. then a second panel can also be cut on the side to accomadate the other half of the tree and attched down the rest of the run fencing... does that make sense? O how I wish I was there to help!!! I'm actually kind of jealous...
O, and bonus of creating a dome, I would think the vine would eventually also use the dome to continue growing/climbing... eventually you will have an arbor of vines.