Do you have an Uber sound system? Or some other HUGE power drawing device? Unless you do, the 105 amp alternator is NOT your problem. Think how many Mustangs were built, lived, and died all left the factory with a 105 amp alternator. Most had no issues with flashing dash lights or the battery not charging.
I can almost promise you there's a reason for your problems. However, since you are certain that the "usual" causes check out fine, this means the problem is likely more esoteric.
Do you have a volt-Ohm meter (VOM) are are you willing to use it? If so one of the tests is to confirm the ground between the alternator case and the battery negative. Need two readings. With the motor running, measure the voltage between the alternator case and the battery negative. Post.
Now measure the voltage at the battery. Post. Even better is to repeat the measurement when the dash battery warning light is on.
Next with the motor off measure the Ohms between the alternator case and battery negative. Should be low. Be sure to zero the VOM meter before taking the measurement and use fresh batteries in the VOM.
Another possibility is a bad ground behind the center console. This ground is shared by multiple devices including the cluster and radio. This ground is frequently messed up during a radio install.
Yet another possibility is a cluster problem. IE, the dash warning light is false. For this we really need to know the voltage at the battery.