1. Measure the room,,length , width & height.
2. Multiply L x W x H to determine the total cubic footage in the room.

Bathroom should have four complete air exchanges per hour so multiply the total cubic footage by 4 to determine total cubic footage per hour.

divide total cubic footage per hour by 60 to determine cubic foot per minute.

example:
L 10' w 10' H 8'

10'x10'x8'=800cu.ft

800cu.ft x 4air exchanges per hour = 3200cu.ft per hour.

3200CFH / 60min per hour = 53.33cfm

I assume it isn't the strength of the fan, it is the vent line. The fan will exhaust 180cfm provided it is vented as intended. If you put in a reducer, you reduce the air flow. The flow will certainly be less than 180, but you are right that it will be over 55.

As a lower limit, assume that 100% of the air that was flowing outside the 6" reducer will be stopped. Dropping from 8 to 6 cuts down the cross section of pipe to 56%. That would give a lower limit of 101 cfm. Of course, the air doesn't all just bounce back. It creates a line of higher pressure that increases the flow through the reducer above that lower limit.

Not sure you are going to win an argument with the inspector though.

Even taking into account the Bernouili Principle, as well as the Coanda Effect, the air actually exhausted exceeds the minimum code requirements in this area, or most others for that matter. But without the expensive equipment, such as a balometer or velometer, I have no means of proving this to the inspector, most of whom are barred from using anything such as commen sense. I would like to find an easily understandable spread-sheet or formula that I could show the inspector, and maybe educate him as to the potential of a particular sized fan delivery at a particular SP considering any/all duct loss will comply with all existing codes.