Is it necessary to attempt to obtain a donated mushaf for Qur’an memorization centers and remove paper scraps with the name of Allah written on them, given the difficulty of doing so and the impending, irreversible travel?

You are not required to try to reach the mentioned Mus'haf to remove the papers from it, even if it were easy to do so. For the religion of Allah is ease, and hardship is lifted, as Allah Almighty said: "And He has not placed upon you in the religion any difficulty."