This is not a complete answer.

You can find some information on the "relative version of Drinfeld's double construction" in Lurie's DAG VI: E[k]-algebras.  The explicit construction isn't given for your example, but there is a general description of the universal property characterizing the E[2]-centralizer of a monoidal category with respect to a monoidal functor.  Search for "commutant".

(You may have received some kind of secret document distributed by email before Talbot 2008 describing the $U_q(n)$ construction.)