If you have a copy of Gaitsgory's <i>Notes on factorizable sheaves</i> there is a sketchy exposition of the relative double in section 3.2, and a description of the construction of the quantum group in section 5 (in the numbering of the March 2008 edition).  Here is what I understand, with many possible misconceptions. There is a sequence of Hopf algebra homomorphisms:

${}^{free}U_q^\pm \twoheadrightarrow {}^{DK}U_q^\pm \twoheadrightarrow u_q^\pm \hookrightarrow {}^LU_q^\pm \hookrightarrow {}^{cofree}U_q^\pm$

${}^{free}U_q^+$ is the free associative algebra generated by the simple root operators $E_i$, which have the appropriate grading.  It has the coproduct you described.  Taking a quotient by Serre relations yields the first arrow.  The operation of taking dual Hopf algebra with opposite comultiplication flips the direction of the diagram and changes the sign.  Taking the double yields hybrid algebras, e.g., half Lusztig and half DeConcini-Kac.  I don't understand why both $Vect^\Lambda_q$ and $Vect^\Lambda_{q^{-1}}$ play a role here.

The relative version of Drinfeld's double construction can be written on the category level as the E[2]-centralizer of the monoidal functor $Vect^\Lambda \to Rep ({}^{\ast}U_q^\pm)$, where $\ast$ indicates one of the choices above.  I guess you can combine definition 2.5.1 with example 2.5.15 in Lurie's DAG VI: E[k]-algebras to get a description of this construction.  Generically this seems to yield the category of representations of Lusztig's U-dot algebra instead of the original $U_q$.