Hi Suhdsy,
I'm not allergic to black rubber compounds, but do react strongly to the manufacturing chemicals for neoprene, so some of my experience in prepping gear may be helpful in treating your new sharkskin suit for best results. I generally wind up with full body hives and breathing issues if I don't prep my gear properly. It probably seems like overkill to anyone that hasn't reacted to their gear, but it works for me. In case some of your reaction is due to lingering processing chemicals, this could help.
tldr: use a mild dish soap to wash, soak, and rinse your gear for several days before using.
The Details
Any new exposure suit goes in the bathtub with a safe dish soap for multiple soaks and washes. I use Dishmate Pear soap because I don't react to it and it doesn't leave a fragrance. If you are ok with the original Dawn dish soap, it has a good reputation.
The bathtub, washer or large tub will become soak central for a few days. Start with puting in the suit in tepid water and squirting a medium amount of soap on it. Wear safe gloves or have someone without the allergy work the soap through the suit and leave it to soak. In the evening, give it a long rinse and restart a soap and soak. In the morning, empty the tub and rinse the suit in the shower with you and then refill the tub. Repeat this process till you can hold the wet gear without any issues and can put an arm or foot in the soak water for several min. without getting itchy. (I react quickly, so it is easy for me to figure out when my gear is clean -- YMMV.) Once you reach this point, switch to soaking the gear in clear water and doing long rinses -- for a wetsuit, you can gently walk on the suit in the tub to cycle fresh water through the suit cells. Generally, use one day of soaking/rinsing in fresh water works all the soap out.