Is it permissible to name one's son Abu Bakr Siddique, where Siddique is the middle name?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

There is no harm in Islamic Shariah to name your son Abu Bakr Al-Siddiq.  This is one of the best names.  The names of Prophets and companions, like Abdallah, Abdulrrahman, etc., are the best names.

The Prophet (Sallallahu Alaihi wa Sallam) ordered us to give good names (to our children) when he said: “On the Day of Judgement you will be called by your names and by the names of your fathers, so give good names.” [Reported by Ahmad and Ashab Al-Sunan]

And there is no doubt that Abu Bakr Al-Siddiq is one of the best names as already mentioned.

Allah knows best.