Ah so the military is less likely to hassle you if you're travelling with a dog? That's interesting. We don't have military checkpoints here, but in some places the police will stop people at certain times to check if they have been drinking. They use this as an opportunity to harass drivers and pressure them into searches, and look for other contraband etc. I don't know how that would play out with a large dog, but it might work the same way.
But the police are notorious for shooting people's dogs when they are attempting to search/arrest someone at a residence. Naturally the dog is defensive of his own home, and that's all they need. Any excuse where the officer feels "threatened", and they'll shoot dogs and even unarmed people. Sometimes they are cleared of wrongdoing too because of qualified immunity or something like that.