What should I do to attain the reward of the Prophet's حديث (Hadith): "Whoever prays Fajr (morning prayer) in congregation, then sits remembering Allah until the sun rises, then prays two rak'ahs, will have a reward like that of a Hajj and an 'Umrah," while repeatedly doing so without causing embarrassment to the mosque's muezzin or feeling ostentatious?

This act that you are doing is an emulation of the Sunnah of the Messenger of Allah, peace and blessings be upon him, for he used to sit in his prayer place after Fajr until the sun rose. It is mentioned in the Hadith that whoever prays Fajr in congregation, then sits remembering Allah until the sun rises, and then prays two rak'ahs, will have a reward like that of a complete Hajj and Umrah. As for your apprehension that this inconveniences the muezzin, it might be a delusion. If the muezzin needs to leave before sunrise, you can agree that you will close the door when you depart. Your feeling of showing off (riya') is a trick of Satan. What is required is sincerity and repelling these thoughts. Do not abandon this good deed out of fear of people. Some of the Salaf said: "Performing an action for the sake of people is riya', and abandoning an action out of fear of people is shirk."