The book of Leviticus is entirely a priestly source, filled with rules to live in the land of Israel and rituals.
After Inheriting the Land (after 1428 BC)
Exod 34:22 “And you shall observe the Feast of Weeks, of the firstfruits of wheat harvest, and the Feast of Ingathering at the year’s end.
Lev 23:34 “Speak to the children of Israel, saying: ‘The fifteenth day of this seventh month shall be the Feast of Tabernacles for seven days to the LORD.
The feast of Ingathering occurs at the year end in Egyptian calendar based on the seasonal cycles of river Nile, which was used by Israelites who just came out of Egypt. However, when the children of Israel entered the promised land, the feast of Ingathering or completion of harvest occurred on the seventh month as per the seasonal cycles in Israel. This was discussed in detail in the post How did year end became seventh month? and reading it helps to understand much better. The book of Leviticus does not refer to the feast as Ingathering or does not mention the feast of Tabernacles at year end.
Deut 16:13 “You shall observe the Feast of Tabernacles seven days, when you have gathered from your threshing floor and from your winepress.
It seems that Deuteronomy bridges the gap between Ingathering in Exodus at year end to Tabernacles in Leviticus at seventh month by saying, feast of Tabernacles after gathering from threshing floor and winepress. In other words, Exodus was written first using Egyptian calendar, Deuteronomy transitions from Ingathering to Tabernacle but does know which month and Leviticus nails down the month as 7th. Without having an actual harvest based on seasons in ancient Israel, it is impossible to determine which month will the harvest occur for the feast tabernacles. The seventh month clearly suggests that the book of Leviticus was written or revised after the children of Israel inherited the land, which must be after 1428 BC.
Second year of Exodus (after 1467 BC)
Lev 1:1 Now the LORD called to Moses, and spoke to him from the tabernacle of meeting, saying,
Lev 27:34 These are the commandments which the LORD commanded Moses for the children of Israel on Mount Sinai.
The book of Exodus begins with God speaking from the tabernacle of meeting and ends with God giving commandments to Moses from Mount Sinai. This means, the book must be written as early as the second year of exodus when the tabernacle of meeting was complete.
Based on what we had seen, the book of Leviticus was written sometime after 1467 BC but later revisions and clarifications regarding the feasts days can be clearly noticed. It is certainly possible God might have given Moses to celebrate on the seventh month even before Moses or any Isralite harvested the land, but lack of such information on either Exodus or Deuteronomy clearly points out a later scribe clarifying the details of the month as seventh to regularise it.