I can definitely confirm what others have said about sediment causing skin issues. I usually get skin issues in the form of the itchy/peeling "kava dermopathy" but I wouldn't doubt that it could also cause hives and other dermatological issues.
Like shakas said, I find that certain strains/varieties give me skin issues while others don't. I'm not sure about the cause of this since all kava should be devoid of pesticides, microbes, etc. That being said, there are differences not only in kavalactones, but also a wide array of other compounds found in plants (eg. secondary metabolites like flavonoids) that could be the cause.
The problem is that there are a tremendous amount of causes that would be hard to narrow down. If it's something you haven't experienced before starting kava, then there is definitely a greater probability that it isn't due to something like diet, weather or allergens. Still, I think that at least trying an H1 antagonist like benadryl might be helpful. H2 antagonists like ranitidine (avoid cimetidine) would be worth a shot, though there is less evidence showing beneficial effects from H2 antagonists. Also, be aware that you shouldn't take either within a few hours of having kava (due to sedation, enzyme inhibition and other potential interactions).