Is the name "Abd Al-Mawjoud" permissible, knowing that "Al-Mawjoud" is not one of the Beautiful Names of Allah?

There is no name for Allah Almighty in the Quran and Sunnah that matches the name mentioned in the question ("Al-Mawjood" - The Existent). The names of Allah are a matter of divine revelation (Tawqifiyyah), and it is not permissible for anyone to name Allah with anything other than what He has named Himself, or what His Messenger, peace and blessings be upon him, has named Him.