In the stage of exacerbation, no, no, no, and again no.
In the stage of remission Also not. Firstly, to dyes (especially black and red) used for tattoos, an allergic reaction develops quite often in perfectly healthy people, not to mention people with skin peculiarities in the form of atopic dermatitis. And it doesn't go away until the dye is removed from the skin. Secondly, atopic dermatitis is characterized by the Kebner reaction - an exacerbation of the disease in response to any mechanical damage to the skin. Thirdly, atopic dermatitis tends to worsen in stressful situations, and tattooing is nothing more than a high painful and stressful irritant.
Yes, there is a possibility that no reaction will follow. But is it worth the candle ... to decide for you.
You can, he's not contagious. Wait for a stable remission, stock up on emollients for the remains of an unbroken carcass so as not to provoke a relapse, and after praying alone, go to surrender into the hands of a specialist. Good luck!