I believe honeysuckle is a bit like trumpet vine to kill. You may have to be a bit careful with this method depending on how close the base is to your berry plants.
Cut the honeysuckle down the the base right against the ground. Boil water with a bit of vinegar in it. Pour the boiling water on the stump/roots. Make sure to drench it. Repeat every other day for about a week or two and any time you see a sprout. This will kill it within a year.
As far as the branches are concerned: they are right it will re-sprout as it's very opportunistic. If you don't have a way to get it off property, then you could contain it on plastic sheeting or plywood (anything you have laying around). Let it fully die; a year would be best. Then you could
compost it for another length of time. If you see any sprouts during that time period by all means do NOT place it back into your plant areas. I've never stayed anywhere a very long time, so I'm not sure what happens to it after the one year period. I let mine sit for that time and it looked pretty dead. Maybe someone else has experience with composting the dead matter and seeing if it starts to come back, or if it's truly dead.
Good luck!