It sounds like you're actually allergic to milk in addition to being lactose intolerant. They're different things. Blood tests for food allergies are notoriously unreliable, but if you're actually having asthma attacks, well there you are. The bloating is likely from lactose intolerance.

But, as several people pointed out, mayonnaise doesn't contain any milk. Eggs, oil, vinegar, possibly lemon juice, salt. That's it.

Food allergies rarely come singly. Susceptibility to allergies comes from a hyperreactive immune system (which can have a lot of causes), but actual allergies develop through exposure. Maybe you're allergic to eggs? Or yeast (vinegar)?

Nothing with mayonnaise (or any other raw egg product) in it should be kept unrefrigerated though - everyone's correct about that. I'm very lackadaisical about bacteria, but I draw the line at unrefrigerated mayo...