Is writing "Dedicated to Allah Almighty as a gift from Dr. Muhammad Radwan" inside copies of the Qur'an donated to mosques considered showing off and seeking a reputation, or could it be an encouragement for others to follow suit?

There is no harm in displaying acts of obedience if one is safe from showing off (riyaa'). Indeed, the doer may be rewarded if his intention is to encourage others to perform them, and he will receive a reward similar to that of those who follow his example. However, one should not write their name on copies of the Qur'an, as it is sufficient to show the charity or to write "a benefactor" on them to prevent showing off and to repel Satan.