As you point out, the relation between associators and the quasi-triangular structure of $U_q(\mathfrak g)$ (and the related tangles invariants) exists "only" at the Lie algebraic level, not (not yet) at the universal one. Roughly speaking, this is because the twisting which absorb the associator is not $\mathfrak g$-invariant, ie modifies the coalgebra structure, which a priori doesn't make sense at the level of chord diagrams.
So far I know, there is no combinatorial construction of a universal finite type invariant which can avoid associators. But of course it's something people are looking for.
But it turns out that the theory of quantum $R$-matrices is more related to the theory of virtual knotted objects (see this answer of Greg Kuperberg). This is more or less "Bar Natan's dream" that a universal finite type invariant for virtual knotted objects should corresponds somehow to Etingof--Kazhdan quantization of Lie bialgebras.
There is also a baby version of this, which is Alekseev-Enriquez-Torrossian solution of the Kashiwara Vergne conjecture based on associators. It turns out that they constructs a kind of universal twist which can "kill" the associator, and a kind of universal solution of the quantum Yang-Baxter equation, living in a bigger algebra than the algebra of horizontal chords diagrams. According again to Bar Natan, this corresponds more or less to a universal finite type invariant for "wedded knots". See: http://www.math.toronto.edu/drorbn/papers/WKO/
You may also find this paper interesting : Towards a Diagrammatic Analogue of the Reshetikhin-Turaev Link Invariants