Q.blackberry and other vegetation
How to get rid of blackberry and other vegetation growing between two fences and keep it out. Approx 6″ between fences.

Pulling and digging up the unwanted plants is the natural way to deal with the issue.
If this is not practical, you may want to resort to chemical treatment with a Herbicide.
You will need to follow the directions on the particular product that you use.
https://www.gardeningknowhow.com/plant-problems/weeds/using-herbicide-in-gardens.htm