Bhagavad Gita should be considered as Śruti.
Even though Gita is part of Mahabharata (a Smriti), as you already mentioned it was recited by God himself in form of interactive classified Q&A.
A Shruti is available to the non-audience only when someone remembers it; i.e. converted into Smriti. Hence, if something is eligible for Shruti, then it should be. Otherwise everything will become Smriti! Refer to Distinction between Shruti & Smriti:
Smritis are considered to be human thoughts in response to the
shrutis. Traditionally, all smṛtis are regarded to ultimately be
rooted in or inspired by shrutis.
According to Sanskrit dictionary:
Definition: f. the whole body of sacred tradition or what is remembered by human teachers (in contradistinction to śruti-or what is directly heard or revealed to the ṛṣi-s;in its widest acceptation this use of the term smṛti- includes the 6 vedāṅga-s, the sūtra-s both śrauta-and gṛhya-,the law-books of manu- etc. [see next];the itihāsa-s[ exempli gratia, 'for example' the mahābhārata- and rāmāyaṇa-] , the purāṇa-s and the nītiśāstra-s; iti smṛteḥ-,"accord. to such and such a traditional precept or legal text")
Gita was supposedly retain as it is without any personal opinion from Vyasa. For Arjuna and Sanjaya, it's still categorized as Shruti.