As others have covered, carry a side cutter and be vigilant. I don't know a damn thing about dogs, but I'm sure training it not to fight the pressure isn't too tough. If you've ever set a snare and seen what a coyote does to that thing before it dies, you realize pretty quickly that it's not a quick process and your dog would not be dead before you can get there. Hell, you probably would have plenty of time to run back to the pickup if needed. Most of the time, it seems like it takes the animal wrapping the snare around the anchor a million times until it eventually unintentionally uses the anchor as additional leverage to finish the job.
To be fair to some comments, not all states require stops. Not like that would prevent a death on a dog so not sure what point people are making there anyways. I guess the dog from Instagram wouldn't have had his nose stuck?
One thing that I could maybe see where there is an argument is if the dog ran through a ram power snare. Those things freak me out and to be honest I'm not even sure if there are rules against them on public land; I've never had the intention of using one.
As far as foot traps are concerned since those were also mentioned in the original post, I was part of a foot trap BMP study about a decade ago. Trappers brought in their coyotes/fox and I worked with the wildlife vet to do a very detailed and meticulous dissection tissue by tissue where we recorded even the most faint signs of damage. Everything was put into a database with the trap used and we never knew which trap was used on each animal until after the dissection. The vast majority of people are using offset laminated foot holds. In this case, the most that will happen to whatever canine caught in a foothold is MAYBE some slight bruising. I've seen it duplicated probably hundreds of times with no outliers.