[pmb]

Any computable function f on one variable x has a program. That function is a program of size p. The input x also has a data size d. BB(p+d) >= f(x), by definition, for all f and x. If you think you might have a (function, input) pair (and corresponding (program, data) pair) for which this is not true, see the previous sentence.

[_alternator_]

This approach leaves open the possibility that f(x) = BB(p+d) right?

[linschn]

No, because f is assumed to be computable from the start, which BB is not (otherwise it could be used as a subroutine in a program that solves the halting problem).