It's hard or maybe impossible to figure out what BB(10) is. But abstractly, it is some finite integer. That finite integer is trivially computable: make a turing machine that hardcodes all the digits and outputs them one by one. You can't know that this machine actually produces BB(10) if you don't know what BB(10) is, but that's separate from the question of whether the number itself is computable.

The integer C is computable. However, we do not know what integer C is without first computing BB(10); and doing that might be not be possible.