But BB(10) is also just an integer. So it should be computable?

There is some notational subtletly here. Strictly speaking, BB(n) represents the output of the BB function for the input n. In that sense BB(n) is computable.

However, the BB function itself is not computable. It makes sense to ask "for what inputs is the BB function computable?". When people talk about BB(n) not being computable, they are really talking about the computability of the BB function at n, not the computability of the integer that happens to be BB(n).

BB(10) is computable. That doesn’t mean anyone can actually figure out what it is. It doesn’t even mean it’s theoretically possible to do so. Computable just means that there exists a turing machine that will output the digits.