Roundup is the most cost effective method of controling any type of weed. it can be used as a foliar spray, frill painting ( hack the trunk and spray/paint on) or stump paint.
The best way I have found when controling buckthorn is to do a foliar app in the fall just before leaf-drop. This gives best translocation to the roots and fewest sproutings in spring.
Second is in late spring early summer just after first flush and before sumer drought, cut and stump paint. the plants have expended most of their reserves and are at their weakest.
Last is a dormant cut and stump paint with a 5:1 mix with a hort oil.
From an "green" standpoint I like the stump paint the best because it gives best control of the application. It can be done at any time and even in mid summer I have seen maybe 60% succses. Then in fall go back to the sprouting stumps with a foliar spray or another paint after brush cutting.
The key to sucsess in the stump paint is to get the app done in less the 20 minutes.