All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
It is permissible to give your son the name that you referred to in the question; we have clarified in many Fataawa that it is permissible to give compound names. This is the view we adopt here in Islamweb. Please refer to Fatwa 135831.
Some contemporary scholars disliked naming with compound names as this was not known among the Salaf (righteous predecessors) and because it may cause confusion.
Shaykh Bakr Abu Zayd said: "It is disliked to name with compound names, such as Muhammad Ahmad and Muhammad Sa‘eed, for instance, where Ahmad is the name and Muhammad is for blessing…and so on, as this may lead to doubt and confusion. It is for this reason that these names were not known among the Salaf, and they are among the names given in recent centuries."
The prominent Shaykh Ibn Jibreen was asked: "Are the compound names that start with Muhammad desirable?" He answered: "These names are new and they were not known before. In principle, a name is singular or added to another name, like ‘Abdullaah, ‘Abd Ar-Rahmaan, or Zayn Al-‘Aabideen, and the like. As regards the names that start with Muhammad, like Muhammad Ameen or Muhammad Sa‘eed, there is no basis for such names as far as we know; rather, this occurred among the late-comers so as to seek blessing from the name Muhammad and the like." [End of quote]
For more benefit on the general conditions of naming a newborn child, please refer to Fataawa 82724 and 87262.
Allaah Knows best.