Scott, I believe the source of your confusion is that Majid doesn't claim that the braided Hopf algebra he constructs is both braided commutative and braided co-commutative in C. Just as in the usual case, the Hopf algebra one constructs is braided co-commutative (like U(g)) or braided commutative (like O(G)) if you work dually, but not both. I think your arithmetic about E[n]-operads is correct as to why that would be unusual.
Can you point where in Chapter 9.4 is it claimed that the resulting Hopf algebra is both commutative and co-commutative? For instance in my copy on page 481, he explains U(C) is braided co-commutative but doesn't mention commutative anywhere. On page 477 of my copy, he says "One can use the term 'braided group' more strictly to apply to braided-Hopf algebras which are 'braided-commutative' or 'braided-cocommutative' in some sense." (keyword "or").
By the way, in the Section 3 of http://arxiv.org/abs/0908.3013 is an exposition (not original) reconstructing the algebra A (transmutation of O_q(G)), which uses language probably more familiar to you. Also there is a Remark 3.3 (explained to us by P. Etingof) which gives a more concise description of A (and can be used to derive its key properties like braided-commutativity) by considering module categories and internal homs.
(apologies in advance if i'm incorrect; I read that book awhile ago and opened it up briefly to address your question)