Honeysuckle makes a great green-manure, in my opinion lots of biomass that keeps on coming back... you will have an extremely difficult time eradicating it without uprooting.
I have goats, and they do indeed love to devour it, and if fenced in with it, will strip it completely of leaves. I paid 35 dollars a head for two goats, you can get them even cheaper if you find someone with a billy goat or older whether they dont want to have around any more.
I utilize the feeding force of goats in place of machinery, equipment and manual labor for brush removal on my small permaculture farm in oklahoma. Instead of spending money and time fencing the brush to be removed, just buy 2 or 3 16' cattle panels, wire them together with copper or steel tie wire in a circle, put it right where you need it and move it around whenever you're content with the damage.
Then, to ensure complete eradication, i follow the 'goat tractor' with a 'hog tractor'. Same setup, two hogs i paid 30 dollars a piece for as newly-weaned babies, a couple cattle or hog panels tied together. I feed my hogs on whatever food i can scrounge up dumpster diving a couple nights a week to save on feed.
At first, baby pigs take about a week to completely uproot, aerate and fertilize the area inside 3x16' cattle panels. Now, they are about six months old and can clear the same area in a day or two. They eat the grass roots, and i wouldn't be suprised if they ate the honeysuckle roots as well... and if they dont, they will have them completely uprooted and the soil loosened so you can just grab the remaining roots without any effort whatsoever.
check out the videos i've made illustrating the process for both goats and pigs: