I have a very similar issue with wild grapevine, native morning glory, and peppervine.
The morning glory is the easiest to rip off of whatever it's growing on but, if it's already blooming, pulling it ends up scattering thousands of seeds that tend to germinate as soon as they hit the ground. So I will generally use the propane torch over the area if there's nothing else I want to damage there. That tends to kill most of the seeds and any seedlings that have already sprouted and preparing to take over.
The grapevine & peppervine are a different story. Both are woody perennials and have huge root systems that simply shoot up a new vine 10-20 feet away from the one I cut down. They also grow fast enough to engulf a mature pecan tree in a single season, and the grapevine makes "trunks" that are 6+ inches thick and require a lot of sawing (or a chainsaw) to sever them from the root system.
The best results has been digging/pulling as many of the main roots as possible, as those tend to have the growth buds to put up new growth. Then, it takes a few weeks to monitor the area and dig/torch any new shoots that come up in the area from the roots that broke during the first eradication. I usually spend about 10 minutes each day to monitor the area and deal with any new shoots. Eventually I, either get all of the roots removed, or drain the stored energy the plant needs to send up new growth. But, it takes consistent monitoring, because a couple of days of new growth and photosynthesis seems to accumulate enough energy for it to produce lots of new roots & prolong the battle.
Also, I try to remember the old phrase, "how do you eat an elephant?" ("One bite at a time)."
I found it's best to deal with a small number of plants at a time, instead of taking on the whole system at once. Otherwise I quickly find myself overwhelmed with new growth shooting up a fifty+ feet away. I don't know the growth habit of the honeysuckle, but some of the vines I've been dealing with have been growing, unchecked, longer than the 34 years I've been alive which tends to mean they have a ton of energy stored in the root system.
Good luck with your battle! Persistence always pays off!
