As a precursor - I have worked with four severely dog and human aggressive dogs. Three of them still live with me and the fourth died of cancer. I'm fairly familiar with signs of aggression when they're being displayed. And I didn't see any aggression in that video. I saw a nervous and fearful dog who circled behind his owner and "hid" from the other dogs. Confident dogs don't react that way, and neither do aggressive dogs. His barks were warning barks like "please stay away from me, I don't want you near me, I'm uncomfortable", not "if you come near me I'm going to rip your face off".
If he were truly aggressive, he would have laser-focused on the dog, refused to break eye contact, barked at high volume, snarled, bared his teeth, etc. He didn't do any of that. He's unstable and uncomfortable. Has he ever bitten or attacked another dog?
Ditch the harness. Prong collars on fear aggressive dogs don't help, either. If every time he becomes uncomfortable and reacts he gets pinched around the neck, it's only going to worsen his reactivity. Work on confidence building and BAT/LAT training when dogs are at a distance.
BAT training explanation:
Official Behavior Adjustment Training (BAT) site: humane help for aggression, frustration, and fear in dogs, horses, and other animals.
LAT training explanation (down past where they try to sell you things, they explain the training methodology):
Look at That! A Counterintuitive Approach to Dealing with Reactive Dogs | Dog Training for Dog Lovers Blog
Treating this like offensive aggression is only going to make the problem worse, I think. He's reactive. And fear aggressive if he's actually bitten another dog/person. He needs basic training like how to walk on a leash, how to focus on his handler in public with distractions and how to be disengaged from an uncomfortable situation without trauma or negativity.
