I'm not at all familiar with the plant in question, so keep that in mind before taking my advice.
if you do end up cutting it down, I would recommend doing it during the growing season. that certainly won't guarantee that it won't grow back, but it will take considerably more resources away from the plant than it would if you were to cut during the dormant season when all the sap is in the
roots. if you're fastidious, you'll kill it eventually. you might have to cut regrowth every couple of weeks, though.
you might also consider critters to browse it. goats fenced into the area in question would keep the plants defoliated until they were dead. turn your weed problem into
milk, cheese, meat, and manure. the faster the stuff grows back, the more food the goats get and the more food you get. could take a couple of years to get rid of it all, and you can't count on the goats to stick to just the honeysuckle, so it's sort of a "nuclear option."
I would guess that the woody parts would then make decent
firewood for a
rocket stove or mass heater.