“ʿAṭāʾ used to deny that ʿUthmān had introduced it, saying instead, ‘It was Muʿāwiyah 1 who initiated it. Whichever of the two it was, the practice that existed in the time of the Messenger of Allah ﷺ is dearer to me. Therefore, if a number of muʾadhdhins give the call while the imam is already on the pulpit, and the imam himself gives a call of the type that is made before the muʾadhdhins’ adhān once he has sat upon the pulpit, I regard that as disliked for him, though none of it would invalidate his prayer.’

Similarly, regarding the mosque mentioned in the question, we maintain that it should follow the Sunnah of the Prophet ﷺ, not the Sunnah of ʿUthmān, for two reasons:

First: The adhān there cannot be heard by the inhabitants of the surrounding houses because of the distance, as the question itself indicated; indeed, it is not even audible to those passing along the road that borders the barracks on the eastern and southern sides. In such a situation, adopting ʿUthmān’s additional adhān would fail to realize its purpose.