Why was Shri Rama described as exhausted in the war with Ravana though he learnt the Bala, Atibala Mantras?

Upon departing from the Ayodhya and journeying for 1.5 yojanas, Sage Vishvamitra, Shri Rama and Lakshmana stayed at the southern bank of Sarayu for the night.

It was at that time, Sage Vishwamitra imparted the knowledge of Bala and Atibala to Shri Rama.

गृहाण वत्स सलिलं मा भूत् कालस्य पर्ययः || १-२२-११
मंत्रग्रामं गृहाण त्वं बलामतिबलां तथा |

"Take water, my boy, and like that you receive a group of hymns, bala and atibala from me, and let no time lapse.

न श्रमो न ज्वरो वा ते न रूपस्य विपर्ययः || १-२२-१२
न च सुप्तं प्रमत्तं वा धर्षयिष्यन्ति नैरृताः |

"On receiving these hymns neither tiredness nor fever, nor disfiguring of personality can effect you, nor the demons can charge you either you are sleeping or unvigilant.

—-

Upon continuous repetition of those bala and atibala mantras, the question of Shri Rama getting tired does not arise.

——

During Shri Rama and Ravana war, it was stated that Shri Rama got exhausted.

ततो युद्धपरिश्रान्तं समरे चिन्तया स्थितम् |
रावणं चाग्रतो दृष्ट्वा युद्धाय समुपस्थितम् || ६-१०५-१
दैवतैश्च समागम्य द्रष्टुमभ्यागतो रणम् |
उपगम्याब्रवीद्राममगस्त्यो भगवान् ऋषिः || ६-१०५-२

Approaching Rama standing absorbed in fight in the battle-field, exhausted as he was by the fight, and seeing Ravana who came nearby, facing him duly prepared for the battle, the glorious sage Agastya, who had come together with the gods, to perceive the battle, then spoke as foolows:

———

Hence, the episode of Sage Agastya coming to battlefield and imparting Aditya Hridayam to the exhausted Shri Rama, is an interpolated story, inserted by some later date poet, who composed Aditya Hridayam.