Funny, I was just thinking about this yesterday. People in our area ignore the running at large laws. I was thinking that a paint ball gun might just work. Kazap. Strong enough to hurt at close range but not wound and it would send a warning back to the owner. The problem with a whip is trying to control the horse while you are using it against an aggressive dog might add to the confusion. I can see a wreck happening.
I have an air-soft pistol I pack with me while driving. It's legal and can be purchased without a licence. I drive on paved roads in a small subdivision where I live and also on the back rounds around us (in the woods) and it is there for protection against anything that may threaten myself of my horse - loose dogs, bears, coyotes, people, etc.
We keep a paintball gun in the house for dogs that get into our trash. Sends a nice message back to the owner :) and doesn't hurt the dog in the long term.
I second either the wasp spray or a water gun with some sort of stuff in there like vinegar. Since you'll be driving her, I really wouldn't suggest pepper spray. The little personal cans you get, even in the stream type, have a very limited range because there isn't a whole lot of pressure in there. So, for the first couple of uses, they are effective up to about 15 feet...if your aim is good. Every time you use them after that, the range is shorter.
Wasp spray, on the other hand, is designed to be shot over longer distances and it won't completely incapacitate you or your horse if you happen to get it on you. Pepper spray would.
If that isn't good enough, I'd suggest a bb pistol. They are not usually fatal but they would certainly give the dog something to think about.