Somehow, I doubt that. Especially, if they're heavy grass eaters. Some dogs are more than others. (And they don't do it only because they're sick.)
I use some heavy duty line and don't "floor it" when I weed eat around a chain link. With some careful weed eating, you won't chew up too much line. It helps to accept it may not be perfect, too. I have dogs, as well, and they really enjoy grazing, especially the taller grass along the fence (I have a wooden privacy now, but used to have chain link) so I refuse to use any chemicals. My dogs got really sick after the fuckin' apt complex behind my old duplex decided to use poison along our back fences without any goddamn permission or any informing of the residences. Not even the apt complex's property!!! I had no idea until all the dogs got sick and I noticed all my fucking grass dying along the back fence.
Anyway, no to the goddamn poisons. I wouldn't spray that shit on something and then feed my face or my loves ones. I'm sure not taking a chance spraying it in my yard for my dogs to consume. Just because a human or animal doesn't get immediately ill doesn't mean they're not being slowly poisoned.