The Busy Beaver Game is not a single Busy Beaver, but the game where the objective is to locate the longest-running program with a given program size. What makes it non-computible is largely the fact that how long the program can run for is not known in advance.
If there is a cap on both the program size and the execution of any individual Busy Beaver, then it becomes computable by the trivial expedient of generating every single possible turing machine of the target size, executing each (stopping at the time limit), then returning the one that ran for the greatest number of steps and terminated.
In other words, the Busy Beaver Game is noncomputable because of the Halting Problem, and the Halting Problem is noncomputable because it lacks an upper bound in time.
Incidentally, bounding time also bounds output space because at each time unit at most one unit of output may be written.
> In other words, the Busy Beaver Game is noncomputable because of the Halting Problem, and the Halting Problem is noncomputable because it lacks an upper bound in time.
Not completely. The halting problem is computable for some more limited forms of computation, even if they have no upper bound in time.
In an alternative formulation: for some Turing machines we can prove in finite time that they run forever. Thus your 'lack of an upper bound in time' is not enough.
I believe there exist some subset of turing machines that can be proven to run forever.
But providing an oracle for the halting problem - of which an upper bound is one form - makes the Busy Beaver Game computable.
To get back to the original topic... a finite brain in finite time can only produce finite output. Both the brain and its output are, like all fully bounded sets, computable.
If there is a cap on both the program size and the execution of any individual Busy Beaver, then it becomes computable by the trivial expedient of generating every single possible turing machine of the target size, executing each (stopping at the time limit), then returning the one that ran for the greatest number of steps and terminated.
In other words, the Busy Beaver Game is noncomputable because of the Halting Problem, and the Halting Problem is noncomputable because it lacks an upper bound in time.
Incidentally, bounding time also bounds output space because at each time unit at most one unit of output may be written.