Scriptures say Balaram the older brother of Krishna departed this world whilst in a mediative state and the white spirit snake left through his mouth, and not through Brahmarandhra. Does that mean he will be reincarnated and if so is there any evidence of this reincarnation ?
Balaram is also in the same category of Lord Krishna himself. He takes avataras, doesn't reincarnate like ordinary souls.
For eg: Krishna and Balaram are called two lords of the universe.
athānusmṛtya viprās te anvatapyan kṛtāgasaḥ yad viśveśvarayor yācñām ahanma nṛ-viḍambayoḥ
The brāhmaṇas then came to their senses and began to feel great remorse. They thought, “We have sinned, for we have denied the request of the two Lords of the universe, who are engaged in spreading human pastimes.” SB 10.23.38
tau rejatū raṅga-gatau mahā-bhujau:
the two mighty-armed Lords shone splendidly in the arena. (SB 10.43.19)
Another place we see,
dhvaja-vajrāṅkuśāmbhojaiś cihnitair aṅghribhir vrajam śobhayantau mahātmānau
Two exalted personalities beautified the cow pasture with the impressions of their feet, which bore the marks of the flag, lightning bolt, elephant goad and lotus. SB 10.38.30
Thus we see Balaram has similar qualities like Krishna not of Prthu and other avesa avatars.
His glory is further described.
naitac citraṁ bhagavati hy anante jagad-īśvare ota-protam idaṁ yasmiṁs tantuṣv aṅga yathā paṭaḥ
My dear Parīkṣit! That Balarāma killed Dhenukāsura is not such a wonderful thing, considering that he is Bhagavān, the unlimited and the controller of the entire universe. In him, the entire cosmos rests as cloth depends on its threads. SB 10.15.35
Yamunadevi speaks to Balarama:
rāma rāma mahā-bāho na jāne tava vikramam yasyaikāṁśena vidhṛtā jagatī jagataḥ pate
Rāma, Rāma, O mighty-armed one! I cannot know your full prowess. With a single portion of yourself you hold up the universe, O Lord of the universe. SB 10.65.28
Sridhara Svami comments on the above verse:
"With a single portion," namely Sesha.
yuvāṁ na naḥ sutau sākṣāt pradhāna-puruṣeśvarau:
you are not our sons but the very Lords of both material nature and its creator. (SB 10.85.18)
Balaram doesn't reincarnate as normal jivas reincarnate. He takes avataras.
The word avatara is explained in Laghu Bhagavatamrita:
tal-lakṣaṇam – pūrvoktā viśva-kāryārthaṁ apūrvā iva cet svayam | dvārāntareṇa vāviḥsyur avatārās tadā smṛtāḥ ||2||
If the forms previously described (svayaṁ-rūpa, tad-ekātma or āveśa) appear in this world as ever-unique, either directly or through an agent, for certain purposes in the material world, they are known as avatāras. LB 1.2.2
Baladeva Vidyabhusana comments as follows:
If (cet) the forms previously described such as svayaṁ-rūpa (pūrvoktā) appear (āviḥsyuḥ) in this world directly (svayam) or through an agent (dvārāntareṇa) they are considered avatāras. The form that descends from the spiritual world to the material world is called an avatāra. Matsya and Haṁśa, who appeared directly for Brahmā, manifested forms without a visible source, according to Mahābhārata and other scriptures, whereas Garbhodakaśāyī-viṣṇu appears from Kāraṇodakaśāyī-viṣṇu. Kṛṣṇa appears from Vasudeva, and Rāma appears through Daśaratha. What is their purpose? They appear to produce the universe or to perform activities within the universe (viśva-kāryārtham). That means they appear to produce mahat-tattva by agitating prakṛti (Mahāviṣṇu) or to satisfy the devatās by killing the sinful, to give bliss to the devotees longing to see the Lord by becoming visible to them, and to preach pure bhakti. Apūrvā iva means that they appear ever-unique, as if not having appeared before. Thus they are all remarkable.