Add a rotary axis B on a lathe

I guess it kind of depends what that B axis actually does. How would that axis interact with the XZ and or the spindle?

So is C a spindle and an axis as in a lathe with a spindle that can be positioned or is the B the only spindle and the C just a rotary axis? To me this looks more like a 4 axis XYZA milling setup with a rotary axis A and without the linear Y axis.

Ok then your machine is milling machine with a horizontal spindle and a rotary A axis without a Y axis:
your X is the Z,
your Z is the X,
your C is the A,
and the Y does not exist.
In many CAM software you can define the rotary A as substitute for the missing Y so the tool path for Y is wound around the A axis. I have no experience with Fusion360 so I can't comment on how to do it there but I would expect it be able to do that.
The post processor would be the same as for a regular 4 axis linuxcnc mill.

so you might want to change the title of the post to get more information regarding 4 axis milling in fusion360