I'm personally quite leery of essential oil use on pets, because they can't tell us if the smell is bothering them, or their skin is feeling irritated, or if they're feeling off after using them. Until they have really severe symptoms and you're rushing them to the emergency vet, anyway - peppermint oil and other mint oil is often listed specifically as toxic to animals. So my first question, before whether it works or not, would be how concentrated this spray is and how likely it is to be absorbed into the skin or licked by either dog in the scenario.
Personally, as the owner of a dog who does get this kind of negative attention (neutered males are rare here), I just don't bring him to dog-social events. He only socializes with our other dog and a select few known dogs (mostly owned by family) that he gets along with. It's not the only reason - he also has reactivity and arousal issues around strange dogs that can make him quite the jerk and create unsafe conditions for everyone - but it's what I've landed on after spending many of his early years taking him to dog parks and doggy day care. Just not worth it.
Personally, as the owner of a dog who does get this kind of negative attention (neutered males are rare here), I just don't bring him to dog-social events. He only socializes with our other dog and a select few known dogs (mostly owned by family) that he gets along with. It's not the only reason - he also has reactivity and arousal issues around strange dogs that can make him quite the jerk and create unsafe conditions for everyone - but it's what I've landed on after spending many of his early years taking him to dog parks and doggy day care. Just not worth it.