Personal curiosity question for you gentlemen that deal more often with chair prone trees (not necessarily talking about the currently pictured tree as I know its lean is severe...):
If you are trying to swing a chair prone tree, would putting in a soft dutchman help more so than other more pliable trees? Or would it cause more trouble...
There's no way to fall something else on it. Can't get a bucket truck back there, so the only way I can think of to get the limb off is to hire a climber to do it. There's one other time consuming but fun trick I could try. I've done this a number of times, and everybody is probably going to laugh at me about it. I put together a mason's scaffold under the limb, and top the scaffold off with a platform consisting of a piece of 4x8 plywood. Two tiers will get me 12 feet up on the platform. I place another tier of saffolding above the level of the plywood platform, these serve as guard rails. Then I tie the contraption down with guy ropes so it won't rock while I'm on it. Then I use a pole saw to cut the limbs up into small pieces and drop them. I don't cut heavy or really long sections due to the risk of one swinging back at me. The thicker sections get cut into three foot chunks and will drop with no problems. Then I slice 'em in half for firewood.
I've done my share of construction work on scaffolds, so I'm used to working on them and assembling them. This tree has pickup truck access, so this non-professional method of mine is doable if the scaffold measures out to get me within working distance. If I got to go up to an 18 foot platform, then I ain't doing it and I'm back to square one.
Chuckwood- I would have to see the tree in person, but it looks like you would have a tough time at it with just a saw. I've swung some ugly crap like that pretty far, but its kind of a crap shoot. I don't really know how well the holding wood is in a hackberry. If you've got a good root to use to pull it around you may get it to go. I'd dutch off the entire far side. You will need a wedge in the back corner to keep the cut open so you can cut enough off. Then start cutting from the swing side to the pull side. If a tree is already half cut off the stump it won't chair. Hit the wedge to nudge it forward if you can. Nip a little bit of wood, etc. If the wood holds you (I) could probably get it there. Leave A LOT of holding wood and nip at it. Watch the top, nip, watch, nip. Hit wedge. If its starts moving the right way, let it. Swinging chair prone trees is not much different from swing any other tree. If half the tree is cut off before anything happens (movement) it won't chair. It might tear out and wreck the log (not an issue in your case), but it won't chair. In all honesty If the fence is important to you I wouldn't do it. I've gotten trees like that to go pretty far, but like I said it can be a crapshoot.