4 legged mules

I have started a few Mules and Horses that were goosy about dogs and if they were bad enough to be dangerous for the dogs "might kick one" I'd hobble all 4 feet with the horse or mule standing in an open area, not tied up, and thoroughly with plastic feed sacks, buckets, tires, barrels and anything else desensitize them, if they dodge and fall get them up and do it some more until they will not move or dodge. Next I have an old semi tire and rim that I'd hook to the saddle horn and ride them down the gravel road pulling it, it is noisy and scary looking, if they want to run let them, you can probably ride farther than they can run, when they get tired and want to stop running make them go some more. Then continue pulling it at a walk if you are pulling it on the right be sure and turn them to the left so the rope gets against their thigh and or hip and do the same from the left, make them face the tire and walk up to it until they are no longer afraid of it. The first hunt or two don't take them fresh, do a really good job of having the edge off them and maybe have given them a refresher course of desensitizing. Most do not require that and you can just pay attention to their behavior and keep them from responding severely to the dogs running up to them and it is just OJT. Have only a very few times allowed a dog to ride with me and then only if it was hurt too bad to travel or keep up on its own, then I have tied my mule real short and laid the dog beside the mule with a leaswh on them and pull them up in my lap and lean up and untie the mule, or if someone was with me have them hand me the dog. An animal I am hunting on there is always a halter under the bridle with a long lead rope attached.
Hog wise I've never had bad trouble riding to a bay, some horses and mules you will have to spat and spank up to the bay a time or two but not too western. If I'd ever have had one that was real scared of hogs I would have penned them close to some hogs or put some hogs in the trailer and left the horse or mule trapped right next to them a day or two. The first time or two I lead dogs on a mule I do not tie them to the mule and have them on a long enough rope they won't get kicked and you have to pay attention to not let them run under your mule or get the leash under their tail just like if you had a cow roped, pulling tires logs brushy limbs and that sort of thing usually takes any potential problem out of being able to lead your dogs. I have came out of the woods leading dogs on both sides of them mule and dragging 2 hogs with no problem. I can also make my dogs stay with me and follow if need be.
Curious as to why you would want a dog to ride with you?? If all that rigamarow don't work get you another one and run that bad one through the loose horse sale.