So rather than continuing to fight this battle, I decided it was time to remove this plant. The runners it sends out are easily pulled up and re-planted (a definite warning sign), and, unfortunately, I had re-planted these runners to two other beds in this area, so I have a big battle on my hands. I considered three methods for removal: smothering with a layer of mulch, digging out the runners, or the use of a herbicide, such as Round-up. Digging out the roots is the preferred method, but every trace of root must be removed or the plant will re-sprout.
The area I tackled yesterday was in a place that is hard to get to with a shovel and near a rosebush I don't want to inadvertently damage with herbicide. After unsuccessfully trying to remove the underground runners with a handtool, I decided to cover the area with thick, construction-grade plastic and hope that the combination of heat and lack of light will kill the plant. I'm not sure how long it should take for this to happen, and I'm guessing the plant will outlive my patience for having black plastic in my flower bed. So in a few weeks, you may be reading that I've taken up the plastic to find the Mexican petunia, alive and well. If that happens, applying Round-up with a paintbrush is my next move. Gardening is one big experiment.