This is not an answer.
Scott, I am trying to understand the difference between the two. Could you please explain the reason for BB being OK? It seems to me that the usual argument for existence of the values for BB should be provable in a very weak set theory. We form the set of halting TM with n states, prove that it is finite, and take the maximum of steps before halting for each of them. The reason we can not compute the values is the logical complexity of the formula defining BB, we could compute it if it was $\Sigma_1$, but it is not. Am I correct?
I guess that the distinction is about the complexity of the formula defining the function. It seems that you are OK with arbitrary quantifiers over natural numbers but not over sets of them. For example, what would you say if we use GC in place of CH?
So you are asking about arithmetical functions. What about BB for Turing machines with oracles in the arithmetical hierarchy?
Is using higher order quantifiers over natural numbers OK? What if I define it to be the BB for functions defined by such formulas?
I think the relation with truth predicate is that since you are OK with arithmetical formulas, you think they have definite truth values, but it seems that you don't think that formulas outside this hierarchy, e.g. those with set quantifiers over natural number necessary have definite truth values.