This is a great list of the essentials. The only thing that I would add to this list is sutures. If an artery gets cut you will need them to tie the artery off, also for any chest cavity pokes/ gashes. I also carry Benedryl or any antihistamine with me incase a dog has an allergic reaction to something.
I keep Benadryl here at the house but never thought about taking with us.
IMO staples are easier to use on a squirming dog, I carry cat-gut with me but have yet to use it. I always go for the staples and a handful of tampons. I was told by a vet once to just pack a hole/puncture full of gauze and pinch it off, it has always worked.....the tampons came from my uncle and momma who tell of the old timers using them to cut stud horses, they are much easier to use and remove than gauze.
What kind of arteries are you tying off ? I am just curious as I havent seen anything that didnt need to be packed ASAP to stop the bleeding.