I think you have a very short amount of time after cutting down a
tree in which to paint the cut surface with neat glyphosate. (Active constituent in roundup) Like a few minutes or so. If the
tree continues to shoot i'd use a battery drill to drill holes at an angle into the cambium layer, the
growing layer under the bark, and inject straight glyphosate into the holes. Then plug them with a bit of wood or whatever. Drill the holes so they run under the bark not straight into the wood. It won't hurt the surrounding
plants. I bet the stump is a beauty. It'll make a nice feature.