Invasive bush removal tips

You need a folding hand saw to carry in your pocket.
  If you want the job to go faster and easier, I highly recommend this. The shorter the blade, the better. This is because you will jam the saw into the ground to sever stubborn roots that are hindering your progress.  It really saves time because sometimes there is just one big root holding on and its easier to saw it rather than trying to leverage it out. Also you use it to saw off branches that are in your way.
  You will abuse the saw and cheap ones don't last very long. The Felco 600 I sell with the tool is the best and most durable one out there for this job. You can get replacement blades too. 

Use your brain a lot. Not just your strength.
  The idea is to get it out with as little physical effort as possible and the more you plan, the easier it is.
  EXAMPLE: Clear the loose debris around the stump and look closely while you rock the bush back and forth. Look for ground movement that will give you an idea how many support roots it has and where they are.  On inspection, if you see a large root coming from the base, it is often easier to hook under that individual root and sever it from the stump before you hook under the crown to pop the whole bush. Its like you do it in 2 stages by weakening it first, then you deliver the death blow.  Then, when the bush is gone, you pull out or saw off underground the leftover severed roots.

Severed bush honeysuckle roots re-sprout if the cut end is exposed to air and light.

Severed bush honeysuckle roots do not re-sprout if the cut end is 2" or more underground.
  This is a fact some of you may dispute, but I stand behind it from experience. You can even kill a whole honeysuckle stump by covering it with dirt and packing it down good so theres at least 2" of dirt around it. It will not sprout.  I've done it several times. Its awesome!