The main question is right below, if you want to avoid the scriptural reference in the middle. also below we are following Amanta calendar (similar to South India)
Shri Rama was supposed to be crowned as the heir in the month of Chaitra when the moon was in Pushya constellation (VR 2.3). It is very clear that the very next day in the morning Shri Rama is compelled to go for his fourteen year exile.
This means as per Ramayana, the fourteen year period is completed on the 5th tithi and since it started in Chaitra, the ending also should fall on the 5th tithi of Chaitra Shukla (In order for him to return on the 6th tithi).
As per this answer, quoting from the Padma Purana, the Ramayana war lasted till Chaitra Krishna Chaturdashi.
The Padma Purāṇa and Skanda Purāṇa (containing almost the exact same Sanskrit verses) do mention Shri Rama meeting Bharadwaja on the 5th tithi and entering Ayodhya on the 6th tithi, but instead of Chaitra, the month is stated as Vaishakha, (i.e. same tithi as Valmiki but of another month) which is clearly one month beyond the exile period mentioned in Ramayana. (Refer this answer)
If we consider Pushya Nakshatra (the nakshatra when Shri Rama left for the forest as well as when he returned to Ayodhya) to occur on the 6th tithi as mentioned in the Ramayana, then as per basic Jyotisha principles, the month must be Vaishakha (i.e. in line with the Padma Purāṇa). Yet the discrepancy is created because before leaving for the forest the Ramayana mentions the month as Chaitra (as referenced above), which is not accurate, speaking from the Jyotisha perspective (i.e. Pushya Nakshatra on 6th tithi of Chaitra is not accurate).
In further support of the inaccuracy of 6th tithi of Chaitra and Pushya Nakshatra, it would be pertinent to note that Bharata’s birth occurs in Pushya Nakshatra but on the 10th tithi of Chaitra (VR 1.18.15), which is absolutely correct from the Jyotisha perspective. (The 10th tithi of Chaitra and Pushya Nakshatra is correct but the 6th tithi of Chaitra and Pushya Nakshatra is inaccurate)
In the event of such an extension, Bharata would have entered the fire (VR 2.112.25-26). The dates in the Purana look very convincing but yet the Ramayana and Purana apparently don’t match.
Question: Which is the correct date of the above two on which Shri Rama alongwith Devi Sita and Lakshmana returned to Ayodhya? Is it Chaitra Shukla Shashthi or Vaishakha Shukla Shashthi?
Kindly note: this is not a duplicate of Did Lord Rama reach Ayodhya after 20 days from Dusshera? as this seeks a clarification regarding the dispute between two scriptures rather than focusing on disproving Kartika Amavasya (Purnimanta calendar, Diwali) as the return to Ayodhya.