Bermuda grass is the number one weed I deal with down here in North Texas also.
What do you know about the variety of blackberries that you are growing? Do you cut them down or back every year? You may not be able to do anything about the Bermuda now, but once the growing season is over, you can work on it for next year.
If you cut the vines back, then I would at the same time pull out or use a trowel or something to pull all of the tall grass out of the area, then fold newspaper in strips that you can lay on the ground, overlap if you need to, to cover the ground (pave) around your vines. Use strips because I assume you wouldn't cut vines to the ground, you'd still have some stems for new growth to emerge from. Once you have folded paper (and make sure each layer of newspaper has several sheets in it) then I would put a good quality compost or mulch at least a couple inches deep on top of the newspaper. This will at least slow down the Bermuda next year. And if you see it coming through, then you might spot spray with the 10% vinegar.
This is all based on the removal of at least some of your berry vines so you can push the paper through the area from the outer edges, or work directly on top if you remove most of the sharp parts with the annual trimming.
If you don't cut the vines back, you still might be able to drop a lot of mulch down through the existing vines and try to smother the grass. And spot spray next year to prevent it from growing big. Once the vines are woody I doubt the vinegar would hurt them, probably only fertilize them.
Hope this helps. I grew up in the Pacific Northwest where we were trying to get rid of the blackberries, not the grass! (Himalaya blackberries are the kudzu of the Puget Sound region.)