Hi Simon:
This is Kimberly from landscape-solutions-for-you. Thank you for taking the time to ask a question with your ivy problem. Also living in the southeast, and being a landscaper, I have had to deal with this problem with many clients.
I don't believe your tree will fall over if you kill the ivy. The one thing ivy can do is choke out a tree. Your goal is to kill all the root system of the ivy. When my crew and I have dealt with this problem, we would cut all the roots around the base of the tree. We would put down Round Up, or a similar root killer and apply it to all the roots. Over the next two weeks, your ivy will start to die off from the bottom of the tree, and work its way upward. Within in three weeks to a month, your ivy on the tree should be dead. It can be time consuming, but well worth it to remove the ivy.
I hope this helps and if there is anybody out there that has had to deal with an ivy problem, please feel free to give some advice to Simon. Also, if there are other landscapers out there that do something different, please feel free to commit to Simon.
Thank you Simon for visiting this website, and feel free to come back anytime!