Can an Alar Reduction Compromise Breathing?

The short answer is very unlikely. Your nostrils are not the narrow part of your respiratory tract. That part is about an inch back from the nostril opening and is called 'the nasal valve'. Therefore the base of your nose can be narrowed, usually as much as needed without compromising your breathing. If the narrowing is excessive, then you would expect not only trouble breathing but an abnormal appearance.

Alar reduction typically refers to excisions of the nasal sill or nostril excisions to reduce nasal tip width.  This usually does not result in breathing difficulties since the external nasal valve is not compromised by this procedure.

Anything is possible but a correctly performed alar base reduction will doubtful give you any airway difficulty.