I am a veterinarian and I also carry the Fido Pro Airlift.  It gives you solid way toe vac a dog from backcountry with a solid support.  It is not easy, but can be used to carry dog while you also carry pack.  Awkward, but possible.  I have 55 pound Border Collie and have practiced with it.   Glad you teach canine first aid...most people hiking with dogs are not prepared.