One idea which should not be overlooked is that this is the beginning of a Chumash, and besides being part of Torat Moshe, each sefer can stand on its own. Therefore, setting is important.
In Bereishit this is established as the beginning of Creation. In sefer Shmot this is accomplished by repeating a pasuk from sefer Bereishit about those who went down to Egypt: וְאֵלֶּה, שְׁמוֹת בְּנֵי יִשְׂרָאֵל הַבָּאִים, מִצְרָיְמָה: אֵת יַעֲקֹב, אִישׁ וּבֵיתוֹ בָּאוּ. In sefer Vayikra we have וַיִּקְרָא, אֶל-מֹשֶׁה; וַיְדַבֵּר ה אֵלָיו, מֵאֹהֶל מוֹעֵד לֵאמֹר. In Bemidbar we have an elaborate description of time and place: וַיְדַבֵּר יְהוָה אֶל-מֹשֶׁה בְּמִדְבַּר סִינַי, בְּאֹהֶל מוֹעֵד בְּאֶחָד לַחֹדֶשׁ הַשֵּׁנִי בַּשָּׁנָה הַשֵּׁנִית, לְצֵאתָם מֵאֶרֶץ מִצְרַיִם--לֵאמֹר. And in Devarim we have five full pesukim to elaborate upon time and place:
Others (Shadal, Aharon ben Yosef) focus on the role in dating the instruction, and the distinction between Behar Sinai and Bemidbar Sinai.
As the Karaite Aharon ben Yosef writes (right column, to the right) that the words bemidbar Sinai is to distinguish it from behar Sinai. After all, earlier in parshat Behar (from sefer Vayikra) we were told behar Sinai, and so all those parshiyot were from there.
There is a progression from Yetzias Mitzrayim to Har Sinai to being in the wilderness in Sinai. But (now from the super-commentary) since ain mukdam umeuchar baTorah, certain things happened in a different order than as they appear in Mikra. Thus, Shmitta, Yovel, and so on were before the rest of Toras Kohanim even though they appear later in the text. Bemidbar Sinai in the Ohel Moed is after Har Sinai, once the Mishkan is built. (In between, Hashem spoke to Moshe in Moshe's own Ohel.)
Shadal writes something similar:
במדבר סיני: כל מקום שנאמר בהר סיני זהו קודם שהוקם המשכן, אבל משנבנה אוהל מועד לא נאמר אלא במדבר סיני ( רשב"ם ), כי אז לא היה ה' מדבר עם משה בהר, אלא במשכן, בכל מקום מן המדבר שהיו חונים בו ומקימים בו המשכן.
One might bring a related point from Rashi on parshat Behar about mah inyan shmitta eitzel har Sinai:
on Mount Sinai What [special relevance] does the subject of Shemittah [the “release” of fields in the seventh year] have with Mount Sinai? Were not all the commandments stated from Sinai? However, [this teaches us that] just as with Shemittah, its general principles and its finer details were all stated from Sinai, likewise, all of them were stated-their general principles [together with] their finer details-from Sinai. This is what is taught in Torath Kohanim (25:1). [And why is Shemittah used as the example to prove this rule, especially since the very fine details are not even specified here (Sefer Hazikkaron)?] It appears to me that its explanation is as follows: [At the plains of Moab, Moses reiterated the majority of the laws of the Torah to the Israelites before their entry into the land of Israel, this reiteration comprising most of the Book of Deuteronomy. Now,] since we do not find the laws of Shemittah [“release”] of land reiterated on the plains of Moab in Deuteronomy, we learn that its general principles, finer details, and explanations were all stated at Sinai. Scripture states this [phrase] here to teach us that [just as in the case of Shemittah,] every statement [i.e., every commandment] that was conveyed to Moses came from Sinai, [including] their general principles and finer details [and that the commandments delineated in Deuteronomy were merely] repeated and reviewed on the plains of Moab [not originally given there].(It is unclear that others mentioned here besides Rashi would agree with this.)
One might also cast this as the Torah's explicit Documentary Hypothesis. (Not my own theory, but I will not attribute it here online.) The Torah informs us that it was given over in three different locations and times, and we can consider these three texts. There is HS (Har Sinai), OM (Ohel Moed) and AM (Arvos Moav). Each time was a different nevuah, and all were given to Moshe. Moshe may have then compiled these together, and this could perhaps account for what seems to be three textual strains.