You can cut them right down, and dispose of the foliage and dig out the roots, hard work but clean and effective.

If you spray with roundup/or SBK [shrub brush and root] killer and keep the animals away until the spray dries there shouldn't be a problem. Its not usual for them to eat bramble leaves.

I'd go for a combination of these - first cut them down and then treat the new shoots with SBK but it might need more than one application. If the area is suitable regular mowing finishes them off as it leaves the roots nothing to support them, and in that case you might manage without the SBK if you're worried about pets