A mixed equivariant derived category in both the mixed Hodge and $\ell$-adic settings exists. As far as references go, for $\ell$-adic you could use Laszlo-Olsson's papers on six operations for Artin stacks (they should be on Olsson's webpage http://math.berkeley.edu/~molsson/, and probably also on the arXiv). There is no real reference for mixed Hodge modules. However (and this will work in the $\ell$-adic setting too), the Bernstein-Lunts approach works just fine (for linear algebraic groups) even if you are using mixed Hodge modules. The point is that Bernstein-Lunts predominantly just use the 6 functor formalism in their approach, so as long as you have this (which you do for MHM) everything goes through fine. The only place where you might worry is subtleties with weights. However, this isn't a problem if all the morphisms showing up in your approximation spaces are algebraic plus proper or smooth. I remember a couple of years ago going through things carefully for $B\backslash G/B$ to make sure there were no issues. You may want to look at O. Schnurer's thesis where some of these checks are also done. Or just ask Wolfgang!
Let me also add, reading a bit between the lines of your question. As far as graded representation theory type applications goes, you can avoid working with a mixed equivariant derived category most of the time. Usually, the mixed structure is required to prove splitting of some sequences, and/or to get a grading on Ext-spaces. Whenever you need to do this just work on a suitable approximation space and the mixed derived category on this space. Even working in the (non-equivariant) mixed derived category can often be avoided, since usually the only way you have of getting a handle on Ext-spaces is by interpreting them as cohomology of some space or using a suitable `fibre functor' (think Soergel bimodules).
Again from the point of view of gradings in representation theory, the point is that it is the ordinary (i.e, in the non-mixed category) Ext-spaces that one is after. Mixed sheaves and the like offer the opportunity to impose gradings in a functorial way (note: Ext groups in the mixed derived category are just ordinary vector spaces, it is Ext groups in the non-mixed derived category that inherit mixed structures via realization). Getting gradings this way is of course not at all a triviality: a grading this way corresponds to having split Tate structures which can be quite difficult to prove (I do not know how to see directly that Ext between Vermas, in just the ordinary non-equivariant case of $G/B$ are split Tate; mixed Tate is easy but showing everything splits not so much).
In representation theory type settings one always (somewhat mysteriously) has that intersection cohomology always has a basis given by algebraic cycles (think $G/B$ or geometric Satake). As Wolfgang would (probably) say, if we had a fully functional theory of motivic sheaves all these problems with imposing gradings would go away!
A comment on Dan Peterson's answer: I am not sure about a 6 functor formalism for simplicial varieties. There is a brief discussion in Bernstein-Lunts about some of the difficulties in defining six functors and proving their properties in this setting (I don't remember the precise section).
