Well, it's computable as in there is a function that computes BB(n) for a specific n, but you'll never know which function it is.
But the BB(n) function is essentially the Halting Problem which is not computable?
If you want a function to compute BB(n) for any supplied n, then it isn't computable. But a specific number e.g. 123, BB(123) is just a number and for every integer i there is function const_i that computes i. You just will never know which f_i computes BB(123).