Dear civ IV fan 6a,
Find the base of the honeysuckle and whack it. Be careful not to injure the buttonbush as you whack the honeysuckle. When new honeysuckle tendrils emerge from the vine's root system, whack them again. With enough whacking, the honeysuckle will eventually give up and die. By whacking, you avoid poisoning our environment with weed killer.

But once you whack it, you could take a foam paintbrush and dab just a bit of herbicide on the cut stump, especially around the edges. Applied carefully like that you won't hurt the buttonbush and you won't introduce the weed killer anywhere but to the honeysuckle.

Thanks for the tips. I'll wack it and use a q-tip on the exposed stem. I have used that method with some weed trees with great success. I was concerned that since the roots were intertwined with the button bush, that it would kill both.

Honeysuckle is shallow-rooted. You may be able to yank it from the ground and not break a sweat.

Despite the fact that the roots are intertwined, there should be no physical interaction between the two that would cause the chemical to transfer since these are different plants entirely (not the same species/genus).