The Busy Beaver deals with two kinds of programs: those that can use infinite amounts of time and space, and those that only use finite amounts of time (and thus also only finite amounts of space).
As far as I can tell, there's no place in the Busy Beaver ever, where space is finite but time is infinite.
And in any case, if your space is finite, you don't need infinite time: you 'trivially' can detect that you are reaching a tape state that you have reached before and abort. The busy beaver is harder than that.
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.
Your last point is correct, though: if you can detect a "cycle" (like the one the Turing machine I linked to goes through) then you can conclude that the machine won't halt.
Yes, it's trivial to write a Turing machine that only uses finite space but infinite time.
But as you agree, they are of no concern for the computability of Busy Beaver numbers. (However, they are of major concern for people who want to find Busy Beaver numbers in practice.)
An immortal human might be able to produce incomputable reasoning, but I would say it's more reasonable to talk about humans with finite runtime.