Lopping bits of conifer type tree
It looks like a cupressus type of a thing. I've never known them to regrow when cut back to the main stem. I think you will be left with brown dead looking branches and a lot of dry earth in which little else will want to grow. Could you twirl the honeysuckle round and round to hide the worst of it? Do you really like the tree? Could you see yourself replacing it with something fresher?