WCollins
You mention 3km vis for a VFR departure; however it seems clear to me that the conditions in the original question would make a plain-PPL case pretty reckless. It may be legally flyable but it probably would not be navigable.
So we are looking at IMCR or IR. As far as I know an IMCR pilot can do a "VFR" departure in 1800m vis.
I don't know about IR; what is the min vis for a VFR departure if there is no published RVR for that field?