how can I get rid of blackberry vines in my mint garden
Blackberry vines are tall and dense, so are mint but the vines scratch me when I pick mint.
You can cut the blackberry branches down if you don't want them growing in certain spots. But blackberries are so good, you don't want to get rid of them entirely, do you? Between cutting back and propagating new bushes, you can get the bushes to be the size, and in the place, you want.
These articles will tell you more: https://www.gardeningknowhow.com/edible/fruits/blackberries/blackberry-pruning.htm