Lakshmana is Sheshang... the supreme snake or the king of every snake. Even Ram was also hit by Naag Pash & was very near to death? Ram is swayam Shree Hari Vishnu so why could he get hit & injured by Naag Pash?
Ramayana is about Sri Hari taking birth as a human. Ravana's boon from Sri Brahma means he cannot be killed but by a human. While being humans, Sri Rama and Sri Lakshmana were as vulnerable to poison as any other human. Albeit, because of the two vidya's their Guru Viswamitra teaches them, Bala and AtiBala, they are able to survive it longer and were rescued by Garuda who simply answers that they have a deep friendship when Sri Rama thanks him and asks him who he is.
Reference- Book I : Bala Kanda - The Youthful Majesties
Chapter [Sarga] 22