Lineage of Rama has been extensively documented, not only in the Ramayana but also in Mahabaratha & Bhagavatha Purana etc. The Bhagavatha Purana is one of the 18 Purana, which among other things also speaks about the genealogy of various dynasties.
While it is definitely possible to create a fictional family tree for a fictional person, it will not corroborate with the other texts, as they are only fictional.
Kurukshetra War of the Mahabharatha period (around 3100 BCE).
• Ikshvaku – the head of the lineage is mentioned in the Rig Veda verse 10.60.4.4.
• The Rishi, sages, Vishwamitra, Vasishta and Bharadwaja, who figure in the events of the Ramayana period and life of Rama, also find mention in the Rig Veda.