I actually have the same reaction you have when I gut a deer.

My eyes start swelling and I break out in hives.

Now I always wear latex gloves and a long sleeve shirt when I gut one. I have learned over the years how to gut one without getting bloody so my shirt isn't ruined.

When I am finished, I take the gloves off and I wash my hands and face with soap and water. If I start to feel itchy I take a benadryl.