Laser
Personally I don't mind tattoos, they remind of Pacific Island cultures etc. etc.
But I think you can write BA off, I know someone who wanted to become cabin crew, had the tattoos lasered off her arms, leaving some minor scarring, but they still didn't like it. No job!
You might have better luck with other airlines, but I'm afraid I think most of them won't like the tattoos either. Is laser not an option?
As already stated above, passengers like their pilots be well groomed.