How could you "solve" the halting problem if, hypothetically, the busy beaver numbers were "small"?
To follow the strategy you outline, we need to be confident a priori that we have an upper bound on $BB(n)$. More snappily, if $f$ is any function with $f(n)\ge BB(n)$, then from $f$ we can compute the halting problem: given an $n$-state machine, run it for $f(n)$-many steps.

Since the halting problem isn't computable, this means that no such $f$ is computable; this is a "global" phenomenon. No particular value of $BB$ is too large to be used, but rather the growth rate of $BB(n)$ prevents us from having a whole sequence of upper bounds, one for each $n$. As a low-level analogy:


  There is no polynomial function which is always above $exp(n)=2^n$. Of course, this isn't because individual values of $exp$ are too big, but rather because of the overall behavior of the function across all inputs
  .