Most all "brush killers" state on the label they must be applied to foliage to be effective. Very few would be effective if painted on plant stems unless they are specifically formulated for that purpose. To be sure read the label.

I've killed "stray" scrub oak by cutting them down, and painting the stumps with undiluted glyphosate (roundup), which is intended as a foliage spray.

I've had good luck painting the stumps of both wild grape and wild rose bushes. Both look like they are struggling to come back this year, and will probably require a second painting in the fall. In the meantime I will just cut them back a bit to keep them from coming back.

I have not tried injecting. The Nature Conservancy has had luck injecting Roundup into Japanese Knotweed. Try googling to see if anyone's done it with blackberry.

If you want to try the stump-painting method and use a small brush and are careful, you should not get any leaking onto nearby plants.