Praise be to Allah.
The full name of Abu Bakr al-Siddeeq (may Allaah be pleased with him) was ‘Abd-Allaah ibn ‘Uthmaan ibn ‘Amr al-Qurashi. His father’s kunyah was Abu Quhaafah.
The kunyah is a nickname which starts with Abu (father of) or Umm (mother of). It usually includes the name of the oldest son. It may also be just a nickname, as in the example given in the question, since Abu Bakr did not have any child whose name was Bakr. The nickname may be given because of something that is unique to that person, such as Abu Hurayrah (lit. father of the kitten). The Prophet (peace and blessings of Allaah be upon him) gave him this nickname because he would often carry a cat or kitten with him.
Another example is the nickname Abu Turaab (father of dust) which was the kunyah of ‘Ali (may Allaah be pleased with him). The Prophet (peace and blessings of Allaah be upon him) gave him this nickname when he saw him with dust on his back and shoulders.
Therefore there is no reason why you cannot give your child the nickname of Abu Bakr, not as a name.
It is well known that Abu Bakr’s real name was ‘Abd-Allaah. Although it was reported that his name was ‘Ateeq, the correct view is that this was one of his attributes (‘Ateeq = freed, i.e., from the Fire). (Al-‘Istee’aab by Ibn ‘Abd al-Barr, 8/963; al-Isaabah, 4/170-171).
Among his children were: ‘Abd-Allaah, ‘Abd al-Rahmaan, Muhammad, ‘Aa’ishah, Asmaa’ and Umm kalthoom.
If you were to name your son ‘Abd-Allaah – because this is the most beloved of names to Allaah – and give him the nickname of Abu Bakr because it is Sunnah to give young children such nicknames in order to make them feel grown up and responsible, this would be an ideal solution. May Allaah help us and you to do all that is good.