Is it permissible to call his son Takbeer?

There does not seem to be any shar‘i reason why this should not be used as a name. 

Takbeer in Arabic means veneration or enlarging something. See al-Sihaah fi’l-Lughah, 2/105 and al-Qamoos al-Muheet, 1/496. 

The basic principle is that any name that does not carry any bad meaning is permissible to use as a name, but the father should choose the best and most beautiful names for his children. 

Shaykh Bakr Abu Zayd (may Allah have mercy on him) said: It is obligatory for the father to choose a name that is good in both sound and meaning, both from the point of view of sharee‘ah and of Arabic language. So it should be good, sweet on the tongue, pleasant to the ear and carry a noble and good meaning, that is free from anything that Islam indicates is forbidden or disliked.

End quote from Tasmiyat al-Mawlood, p. 13 

Hence what we advise is not giving your son this name, which we do not know of anyone using as a name in previous Muslim eras. 

The good names that are mustahabb, such as the names of Prophets and righteous people, are many and there is no need for strange names. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him) said: It is best for the believer to choose the best of names that reflect servitude to Allah, such as ‘Abd-Allah, ‘Abd al-Rahmaan, ‘Abd al-Malik, and so on, and well-known names such as Saalih, Muhammad and the like. 

End quote from Fataawa Islamiyyah, 4/523 

See also the answer to question no. 7180. 

And Allah knows best.