BB(n) for any particular n is always computable, no exceptions. There simply is no single computable function that can compute BB(n) for every n, but for any particular n there absolutely is a TM that computes it.
Indeed, the value of BB(n) for each n is just a number; and it's easy to construct a TM which outputs some particular number. However, we don't know what those TMs are, for the same reason we don't know what those BB(n) numbers are.
The same applies to more limited settings too, e.g. boolean questions like whether the Collatz conjecture holds: I know that one of the following programs calculates the right answer, but I don't know which:
- PRINT "true"; HALT
- PRINT "false"; HALT
(Perhaps we should also allow `PRINT "independent of the given axioms"; HALT` as well!)
Isn't that what "the function is not computable" is about?
Or is the thesis that the value of BB(748) can't be computed?