This is basically what I do. Minus the diesel.I need to do this. Had an old timer tell me to cut them low, mix diesel and weed killer and paint the cut/stumps so they don’t grow back. He named a specific weed killer but I can’t remember it off the top of my head.
I just use Round Up, but I recently ordered a weed kill called 2,4-D which was recommended by an environmental scientist I met while in Texas. He said it's safer. He didn't have many good things to say about Round Up and its main chemical - glysophate.
Whenever possible I uproot the honeysuckle. You can usually pull out the bushes that have a 1" diameter or less. Those too big to pull I cut, then poison the cut area. It works. In areas that aren't heavily trafficked, I like to cut the stump about 12" from the ground. The next season when it's completely dead you can just kick them over with hardly any effort at all. Much better than having a 1" tall stump you can't really get to and you basically have to wait until it rots.
I'm also considering making one of these for the bigger ones I can't pull out:
Jam it under the base, then leverage back to pop it out if the ground.