Most of the literature on birds stems from a confusion of the distinction between regular string languages and context free string languages with the distinction between grammars with and without recursive rules. The two distinctions are largely orthogonal. It is certainly possible, for example, to define certain regular string languages using recursive grammars, and to define certain context free string languages without using recursive grammars.
When you say that "the result must be a context free grammar" I think what you mean to say is that the string language defined must be a context free (and non-regular) string language. But that does not in any way entail that the only way to recognize the string language is by means of a particular context free grammar.
What I meant by "the result will still have to be a context free grammar" was a context-free grammar that incorporates a counting mechanism, as part of its definition. Something like this, perhaps (in Definite Clause Grammars notation so you can actually run it as a Prolog program):
'S' --> 'A'(N), 'B'(N).
'A'([1]) --> 'A'.
'A'([1|As]) --> 'A', 'A'(As).
'B'([1]) --> 'B'.
'B'([1|Bs]) --> 'B', 'B'(Bs).
'A' --> [a].
'B' --> [b].