One of the best sources for Indian History is Rajatarangini by Kalhana ( 12th Century  ) , Kashmir.

He has listed the Kings of India from Mahabharata period,starting from Gonanda,who was a contemporary of Yudhishtra.

One finds the name of Lava as a king of Kashmir .He is shown as having ruled in Kali Yuga,after the Mahabharata War.

Lava’was a renowned prince of Kashmir. He had a vast and powerful army under him, and probably carried on many wars with his neighbours. It is said of him that the noise of his army made his people sleepless, but lulled his enemies to long sleep (death). He built the town of Lolora which, it is said, contained no less than eighty-four lacs of stone-built houses. Nothing more is said of him than that he bestowed the village of Lovara in Ledari on Brahmanas before his death. He was succeeded by his son Kusheshaya, who was a powerful prince. He bestowed the village of Kuruhara on Brahmanas. (Book I,p.7) (Lava→Lav)’ https://www.jatland.com/home/Rajatarangini_of_Kalhana:Kings_of_Kashmira/Wiki_Editor_Notes#L

Lava was the son of Lord Rama and Lava founded the present day Lahore in Pakistan and there is a Temple for Lava.

To me this seems to an erroneous interpretation.

Or it is a deliberate attempt to discredit Kalhana’s Rajatarangini so that it loses its credibility.This would help secular,liberal historians) to manipulate Indian History. https://en.m.wikipedia.org/wiki/Rajatarangini

Ramayana preceded Mahabharata by one Aeon.Ramayana took place in Tretha Yuga while Mahabharata took place in Dwapara Yuga,which followed Treta Yuga.Dwapara Yuga lasted for 8,64,000 years.Our present Kali Yuga began in the year 3102 BC.

It is probable that the descendants of Lava ruled Kashmir and they could have used their Gotra as Lava and this might have been misinterpreted.

‘Lava and Kushama gotra are descendants of Rama’s son Lava and Kusha. Both these Jat gotras are found in Nagla Magola (नागला मगोला), Soniga Kheda (सोनिगा खेड़ाdistrictBadaun

‘https://www.jatland.com/home/Lava