I actually had this issue with my Arab mare briefly and I seem to have actually solved it. However, I think my situation was a bit different?
My horse has had many bad experiences with dogs. The first time when she was quite young and green. Friends of my grandpa were over and I was riding and their miserable POS dog came after my horse, snarling and snapping at her feet. Bless the dear, I think she was trying to save me because instead of bolting she reared straight up and slammed her feet at the dog (she's never in her life reared again). So that was episode one.
Then when I was at a boarding barn, I got into a tiff with these downright nasty girls who started abusing my mare to get back at me (and actually ended up simultaneously creating a mortal fear of dogs AND strange humans in her). They'd sic their Jack Russell on her just for kicks. I should have beat the snot out of them when I had the chance.
Anyway, after that, dogs were taboo for her. I had to warn people because if a dog came into her pasture, she'd go after it with death in her eyes (ears pinned, teeth barred, loud huffing knoises). Thankfully, she never caught one.
Since I've been at Shay-las, she's improved tenfold. Shay-las mom has three big dogs and one small toy breed dog. In the beginning, she'd go after them. But she's always trusted me, and it worked to my advantage. When she was tied up for saddling and the dogs were frolicking in the yard, I'd make sure to talk to her and stroke her, and get her relaxed. I'd bring the dogs over to pet and sniff with her, all while continuing to make the experience happy for her by having lovely well trained dogs and always talking to her to keep her calm.
Over the last two years of having dogs constantly around (they do not go in the pasture), and nothing bad happening, she's completely mellowed. She'll reach out to sniff them now instead of trying to go after them, and on the occasion they accompany us into the pasture, she just ignores them, even when they run right in front of her.
However, my mare is pretty forgiving by nature. Hopefully, this can maybe help you, but I agree with the others that perhaps it may just be his mentality as I've seen it in horses before. It was never my mares mentality, she grew up with dogs and was fine until she had some really awful experiences with them. She's never been bothered by the goats and cats before, it was always just dogs.
Anyway, good luck!