Would it be correct to call kaal bhairava (and it's various forms) an avatar of shiva Or are they ansh of Rudra like Nandi, Hanuman, Veerbhadra etc ? Should Bhairava be seen as a shivagan or shiva himself?
No, Lord Kala Bhairava is not Lord Shiva himself but a Shivamsha. Though like Lord Veera Bhadra, Lord Rudra n all his activities are also count in the activities of Lord Shiva only like destruction of Daksha yagna was done by Lord Veera Bhadra but is credited to Lord Shiva, Lord Brahma's 5th lying head was cut by Lord Kala Bhairava but is also credited to Lord Shiva only and so on. Prd..
Sri Shiva Maha Puranam 3:8:44-51.:->
Nandīśvara said:- 44. O sage, on hearing the haughty words of the lotus-born Brahmā, Śiva became angry as if out for destruction.
45. Angrily, he created a Puruṣa Bhairava blazing in great brilliance. Then he spoke lovingly.
Śiva said:— 4-6. O Kālabhairava, at the outset this lotus-born Brahmā shall be chastised by you. You shine like god of death, hence you are Kālarāja.
47. You are called Bhairava because you are of terrifying features and you are capable of supporting the universe. Since even Kāla is afraid of you, you are called Kālabhairava.
48. When you are angry you will be suppressing the wicked souls. Hence you will be known everywhere as the suppressor of the wicked.
49. Since you will be devouring the sins of devotees in a trice your name will be famous as sin-eater.
50. O Kālarāja, you will have forever the suzerainty over my city Kāśī, the city of liberation, which is greater than all other cities.
51. You alone will be the chastiser of those who commit sins there. Citragupta will write down their auspicious and inauspicious activities.
I hope this clarifies your queries.