The "bonus question" part seems badly thought out. It seems to assume a particular value for n1. But the requirement is symmetrical for n1 and n2, and for the question to not be completely trivial, n1 and n2 must be different numbers (though that's not quite spelled out in the question). So even if there is a "simplest" answer, you can at least exchange n1 and n2.
And there actually are infinitely many possible answers. There only finitely many possible combinations of values for (BAT(x, N1), BAT(x, N2)). Thus infinitely many primes have the same combination, and thus there are infinitely many possible answers. I doubt all of those have some meaningful connection between BAT(sqrt(n1), 10000) and IBM.
