A 12-inch Lexington would have a short useful life

Would a Lexington-class battlecruiser with 12-inch guns instead of 16-inch guns and better armour make more sense?

No. A 12-inch battery would put Lexington behind contemporary capital-ship armament and give the design a short useful life.

A Lexington-class design with 12-inch guns and better armour is rejected because 16-inch guns were the contemporary capital-ship choice. Navies fitting out capital ships wanted substantially larger weapons than 12-inch guns when they could obtain them. Alaska provides a comparison from a later period, when 12-inch guns likewise did not represent the preferred capital-ship armament. Giving Lexington that smaller battery would therefore shorten its useful service life.