After thinking about this, the only thing I would do is suggest to my neighbor that he get a fairly heavy piece of chain (5/16" or 3/8"), climb up a ladder to the break and chain the top of the tree to the stump in such a way that if the "hinge" were to break, the top of the tree would be supported by the stump and not fall on the fence.
Wait until the land dries out and then let him deal with it with his larger machine.
If you absolutely have to do something about it, keep the safety chain in place and using another chain at least several feet longer than the remaining tree top, attach the chain near the upper end of top of the tree, close to the ground, and drag the top parallel to the fence to a point where it is making about a 30 degree angle with the fence when viewed from overhead. (See attached diagram)
Now, drive a tee post or something similar next to the end of the top top of the tree, and chain the top to the post, so you know it will stay in place when the tree comes down.
If you neighbor can support the weight of the trunk with a big enough loader or tractor, remove the safety chain, cut the hinge, and lower the tree on your side of the fence.
If he can't get there with a big enough machine, you are faced with the issue of how badly you want to save your tree.
In any event you need to attach a guide, maybe a 2x8 or so to the stump high enough up to guide the tree over the fence when it comes down. It is hard to tell, but if you trim the tree carefully near the hinge you may be able to prevent it from hitting your tree when it comes down. If not, I would sacrifice your tree, so that the larger tree doesn't hit it and cause it to fall back over the fence, negating all the work you have done so far...
Anyway, attach the guide, remove the safety chain and cut the hinge, preferably with a pole saw.