Is Honeysuckle the thing with big thorns?
Most places you are allowed to prune back to the fence-line, but if it dis-figures the tree to the point where it de-values the home they are able to take civil action for the loss incurred.
Also same if the pruning kills the tree.
If you remove tresspassing parts including roots and the tree fails you can also be liable for that.
Get 2 bids, one for complete removal and one for a fence line prune then see if you can do a deal with her, pay the portion which would be the prune and she pays the balance to have the tree removed. If you cant agree on that then agree on the fence line prune.
If she wont agree to the fence line prune then you are in trouble coz she might get up ya for the stuff mentioned above. To cover your ass you might have to get an arb report on the value of the tree and a home valuation that hopefully will prove the value of her property to rise if the tree gets removed, you will also need some type of value if she's left with a half tree.
Can any mediators help you? Usually the local authority has some mediation which takes the heat off the situation.
Also, over here there is the Private Nuisance law, where if the tree is an ongoing nuisance which diminishes your enjoyment of your own property then she must remedy the situation or offer compensation. This is a powerful way, the lawyers are writing letters to neighbours to get trees cut down or pruned coz of leaf drop, shading etc and it's working.
So, many intelligent avenues to explore first before you poison that tree and start WWIII with her.
I'm lucky, I've cut down all my neibs trees that peeved me off except for the guy up the back who has banana bushes growing over the fence but above the gazebo roof, he just chops them down when I ask, good neibs are an asset.