For dogs, if legal, a 26 or 31" ASP is my suggestion. Dogs move fast, and batons are good at connecting with fast moving assailants.
For either a baton or pistol, a belt, even a cheap leather belt against bare skin, with any reasonably stable holster will work very well. If open carry is required and you must have heavy clothing on for cold weather, a dedicated gun belt with keepers to a pants belt will be very good, particularly if you put a chicago screw through it.
Either the chicago screwed holster on the gun belt on pants belt, or the gun belt on bare skin will be very stable and comfortable. I've used both extensively.
Pants with belts while jogging is kind of heavy, but I guess I'm an oddball, so it's never bothered me. May not relate much to your needs, but I hope the ideas are useful.