“Hare Rama” refers both to Sri Balarama and Lord Ramacandra:
In this connection we may mention an incident that took place between two of our sannyāsīs while we were preaching the Hare Kṛṣṇa mahā-mantra in Hyderabad. One of them stated that “Hare Rāma” refers to Śrī Balarāma, and the other protested that “Hare Rāma” means Lord Rāma. Ultimately the controversy came to me, and I gave the decision that if someone says that the “Rāma” in “Hare Rāma” is Lord Rāmacandra and someone else says that the “Rāma” in “Hare Rāma” is Śrī Balarāma, both are correct because there is no difference between Śrī Balarāma and Lord Rāma. Here in Śrī Caitanya-caritāmṛta we find that Kṛṣṇadāsa Kavirāja Gosvāmī has stated the same conclusion:
yei yei rūpe jāne, sei tāhā kahe
sakala sambhave kṛṣṇe, kichu mithyā nahe
If someone calls Lord Rāmacandra by the vibration Hare Rāma, understanding it to mean “O Lord Rāmacandra!” he is quite right. Similarly, if one says that Hare Rāma means “O Śrī Balarāma!” he is also right. Those who are aware of the viṣṇu-tattva do not fight over all these details. Sri Caitanya caritamrta Adi lila 5.132
Lord Krsna to Yogamaya: The son of Rohini will also be celebrated as Sankarsana because of being sent from the womb of Devaki to the womb of Rohini. He will be called Rama because of His ability to please all the inhabitants of Gokula, and He will be known as Balabhadra because of His extensive physical strength.
These are some of the reasons why Balarama is known as Sankarsana, Balarama or sometimes Rama. In the maha−mantra
Hare Krsna Hare Krsna Krsna Krsna Hare Hare
Hare Rama Hare Rama Rama Rama Hare Hare
people sometimes object when Rama is accepted as Balarama. But although devotees of Lord Rama may object, they should know that there is no difference between Balarama and Lord Rama. Here Srimad−Bhagavatam clearly states that Balarama is also known as Rama (rameti). Therefore, it is not artificial or us to speak of Lord Balarama as Lord Rama. Jayadeva Gosvami also speaks of three Ramas: Parasurama, Raghupati Rama and Balarama. All of them are Ramas.
– Srimad Bhagavatam 10.2.13