I would recommend checking through the various papers by Cegarra and Remedios (look on the archive) They have done a lot of work in this area, but I am not sure if they have an answer for you question. The Duskin nerve as suggested by David Roberts is related to the bisimplicial approach and the relation is explored in various other papers from Granada, e.g. one by Manolo Bullejos and coauthors.

I would recommend checking through the various papers by Cegarra and Remedios (look on the archive) They have done a lot of work in this area, but I am not sure if they have an answer for your question. The Duskin nerve as suggested by David Roberts is related to the bisimplicial approach and the relation is explored in various other papers from Granada, e.g. one by Manolo Bullejos and coauthors.