I don't think there is any need to annoy the grass while you are killing the blackberries. Since the traditional poison (2,4D) was removed from the market about 30 years ago there have been three common treatments for them: Garlon 600, BrushOff, and Roundup. Roundup (glyphosate) needs to be used in double strength and in my experience is still a marginal performer in this job. The main distinction between the three aside from that, is that Garlon 600 is a specific poison for vine-type ("woody weeds") plants while BrushOff and glyphosate are both broad-spectrum and kill just about everything. None of them allegedly poisons the soil, just the plants. Hence if you wait for green shoots and rapid growth, then apply Garlon, you should be able to continue to grow and develop the surrounding plants. That's the way it worked for me anyway.
In case you care, none of them works quite as well as 2,4D and all cost at least 10 times as much. However I did have to shower and wash my clothes after having the 2,4D running down my back for a couple of hours (same active ingredient as 2,4,5T, and a known cause of dermatitis) while I found much less discomfort with the modern ones.
