In my reading [of Exodus 16] I had failed to catch the apparent anachronism of Aaron placing the jar of manna "before the testimony" (RSV) or "in front of the commandments" (NAB).
The issue might be one of punctuation. The RSV and the NAB and the KJV render verses 33 and 34 of Exodus 16 as two distinct sentences.
RSV:  And Moses said to Aaron, "Take a jar, and put an omer of manna in it, and place it before the LORD, to be kept throughout your generations."  As the LORD commanded Moses, so Aaron placed it before the testimony, to be kept.However, the Douay-Rheims (a 1609 English translation of the Latin Vulgate) has a slightly different structure. Here, the beginning of verse 34 is the conclusion of the sentence in verse 33 (note the comma at the end of verse 33):
NAB:  Moses then told Aaron, "Take an urn and put an omer of manna in it. Then place it before the LORD in safekeeping for your descendants."  So Aaron placed it in front of the commandments for safekeeping, as the LORD had commanded Moses.
KJV:  And Moses said unto Aaron, Take a pot, and put an omer full of manna therein, and lay it up before the LORD, to be kept for your generations.  As the LORD commanded Moses, so Aaron laid it up before the Testimony, to be kept.
DR:  And Moses said: This is the word, which the Lord hath commanded: Fill a gomor of it, and let it be kept unto generations to come hereafter, that they may know the bread, wherewith I fed you in the wilderness, when you were brought forth out of the land of Egypt.  And Moses said to Aaron: Take a vessel, and put manna into it, as much as a gomor can hold: and lay it up before the Lord to keep unto your generations,  As the Lord commanded Moses. And Aaron put it in the tabernacle to be kept.Whatever the punctuation should be, the "issue" can be resolved with the following explanation:
Re-read chapter 16. Note that verses 1-30 deal with events which are happening during the first week when the manna appeared. Now note the tone of verses 31-35. I would propose that these later verses are describing an event that took place later (at or after Sinai, since they expect the existence of the covenant), but they are not at all insinuating that these events actually happened before Sinai at all. I will support my proposal with Scriptural evidence:
In Exodus 16:31, the Hebrew phrase bayith Yisra'el is used for the first time. It literally means "the house of Israel". The phrase ben Yisra'el ("sons/children/people of Israel") is used plenty in chapter 16 (verses 1, 2, 6, 9, 10, 12, 15, 17, and 35). But here for the first time bayith Yisra'el appears in Scripture, in verse 31. Why is the phrase "House of Israel" used this time instead of "sons/children/people of Israel"?
I think we can come to the answer by looking for the next time "House of Israel" is used in Scripture. "House of Israel" appeared first in Exodus 16:31, speaking of them calling the substance "manna". The next time that "House of Israel" appears in Exodus 40:38, the very last verse of the very last chapter of Exodus: "For throughout all their journeys the cloud of the LORD was upon the tabernacle by day, and fire was in it by night, in the sight of all the house of Israel." By Exodus 40, the covenant has been made; the tabernacle and the ark and other elements of worship have been constructed. It is here that the context of Exodus 16:31-35 makes sense. Certainly the Israelites, when they first encountered the stuff, said "man na", but Exodus 16:31 is saying that "manna" is what it was "officially" called by the house of Israel, meaning those who were in covenant with God through Moses. My point is that "House of Israel" is a "covenant name"; it describes the Israelites in their covenant with God. As such, Exodus 16:31 is referring to something at or after the time of Sinai.
The language of Exodus 16:33-34 makes it clear that there was now "the presence of the LORD" and the tablets of the covenant. Exodus 16:35 is even more helpful: it is clearly written after the forty years had ended: "the people of Israel ate the manna forty years, till they came to a habitable land".
So if Exodus 16:35 was written to describe an event that took place much later than Exodus 16:1-30, I would argue that Exodus 16:31-34 are describing a later event as well. They are not "placing" the later event earlier in history than it happened, but they are describing the later event in the context of the rest of the chapter about the manna, and using contextual clues (such as the phrase "House of Israel") to indicate that. To further the point, Exodus 16 is the only place where "manna" is mentioned in the whole book of Exodus, and only in those last verses is the word "manna" (in English) used. It makes sense to have included the "future" of the manna in the same part of the story where it was introduced, especially since it simply never gets mentioned again.