I just found out about the Honeysuckle Popper, a long-handled tool that is used to pry out honeysuckles, root and all. It looks like it works along the same lines as the manual T-Post remover. It's a little pricey but I looked up lots of reviews on different forums and people who get it are very pleased with how easy it is to operate.
It is pricey but some garden groups are now buying them to loan out to people.
I got in trouble on here before for passing on what the official state method of honeysuckle removal is - to cut then spray with ******** (something permies don't use). I love that there's another way and if this tool gets enough press maybe that can become the new state response to the problem rather than advocating people use toxic chemicals - that they have tools in local offices (Farm Bureau or Cooperative Extension) that they can lend or even rent out.