Our place here's lousy with honeysuckle too. Damn stuff is everywhere, including growing up through Wifey's Lilacs. Spray them there and you kill the Lilac too.
I cut the things out with my sawzall and "painted" stubs left behind with diesel.
That stopped it from coming back without damaging the Lilacs.
7mm
I fight sumac around here.
My targeted control is to get a sharp hatchet, and cut just deeper than the bark on a downward angle. Rock the bark out away from the trunk. Repeat around the trunk.
Pour some 40+% glyphosate concentrate in a pop bottle, than carefully pour it into the pockets
you made with the hatchet. In a few weeks, whatever you treated is stone dead.
I know some power line tree trimmers. They use the same glyphosate dumped on the live
Part under the bark on fresh cut stumps. They rarely ever sprout again.
Your diesel would be more forgiving of a spill.