DXF and DGN (and DWG) are really hard to support by open source software like QGIS and GDAL because they are proprietary (secret) formats designed by the vendors of CAD software. DXF is a bit more open I think, but the data model doesn’t fit object based GIS data very well.
It might be quite easy though to simply loop every line in the objects of the QGIS layer and write it as a line to dxf. I guess from there you could use your CAD tools?
About the large files. That is weird, unless you are trying to export the entire country in 1 go.
By the way, there’s no such thing as a “QGIS Cityjson file”. QGIS can import Cityjason data with a plugin.
And hey, why did we switch to English here??