You are discussing about CTDI. If this CDTI contains pitch factor and modulation of tube current - that is the modern status - this dose is merely depending on field of view because CDTI is calculated independant from length of field. If you want to get additional infos you should use DLP (dose length product). I hope I understood your problem and situation correctly.
Maybe I can add a more intuitive reply. It depends on weather the FOV you refer to is a scanning or reconstruction FOV. The latter does not affect the dose since it is executed through reconstruction after the scan is finished. The former, however, involves adjusting the diameter of the radiation field during the scan and this will definitely affect the effective dose to the patient.
Skin dose increases, if larger scan FOV is chosen.
Usually, radiation dose deep at the center of the object is rather stable against the choice of scan FOV. Still, it may slightly increase for larger scan FOV , because of some increase of scatter from shallower regions.
These depends on the detail of bow-tie filter (wedge filter) design.
Also, it depends on the size of object relative to scan FOV.
The above may become clearer if you study the design principle of bow-tie filter (wedge filter).