SB 1.3.22 does not mention the name of Rama or the killing of Ravana at all and so the above translation seems incorrect.
nara-devatvam āpannaḥ sura-kārya-cikīrṣayā
samudra-nigrahādīnicakre vīryāṇy ataḥ param
nara — human being; devatvam — divinity; āpannaḥ — having assumed the form of; sura — the gods; kārya — activities; cikīrṣayā — for the purpose of performing; samudra — the sea; nigraha-ādīni — controlling etc, ; cakre — did perform; vīryāṇi — superhuman prowess; ataḥ param — thereafter
Ref: the link of the sloka provided in the question. I have just changed demi-gods to gods and indian ocean to sea. Sagar does not mean the Indian Ocean and there is no word in our ancient culture as demigod.
The actual translation therefore should be something like this : Then, assuming a divine human form, the Lord appeared for helping the gods and controlling the sea by His superhuman power.
So there is NO mention of the killing of Ravana. Just helping the gods and contolling the sea etc does not repesent Sri Rama.
Moreover, according to the Gaudiya explanation of Bhagavatam, the avatara of treta yuga is red-coloured. They had to make this to make the avatar of kali yellow-coloured, ie Sri Chaitanyadeva. In satya yuga, the avatar was white-complexioned and in black in dwapara. It is true that these four colours have been mentioned in the bhagavata.
Garga Muni confirmed in Srimad Bhagavatam that although the child Krishna was black, He also appears in three other colors -- red, white and yellow. He exhibited His white and red complexions in the Satya and Treta ages respectively.and as per yellow that was used in kali yuga as Chaitanya mahaprabhu.
So the texts of Bhagavata along with the Gaudiya and BBT interpretations make it very confusing. If we accept that Vyasdeva was born before Rama, then we have also to accept that Rama's skin colour was red!