Is there a way to lift 6 functors on constructable sheaves to the dg world?
If I understand correctly your are looking for a dg enhancement of the six operation formalism.
There seem to be a paper that does something very close to it: Yifeng Liu and Weizhe Zheng, Enhanced six operations and base change theorem for sheaves on Artin stacks (available at http://math.columbia.edu/~liuyf/sixi.pdf).
They use the language of $(\infty,1)$-categories, but I think one can adapt it to dg-categories (assuming that one is working over a field of characteristic zero).
EDIT Nov. 27, 2012: the above preprint has been posted on the arXiv: http://arxiv.org/abs/1211.5948