I guess I will try something a little different.
In Webster's dictionary (1828 edition) we get the following definition for the word synagogue:
1. A congregation or assembly of Jews, met for the purpose of worship or the performance of religious rites.
2. The house appropriated to the religious worship of the Jews.
3. The court of the seventy elders among the Jews, called the great synagogue
We can say that the synagogue (at least in terms of the traditional view of a synagogue) didn't exist in 600 BC. This makes the existence of a synagogue (that assumes all of the stuff that goes with it) in 600 BC an anachronism. But, synagogue as a word exists in English, and it appears in the King James Bible. So, we might have two options for the appearance of the term synagogue in the Book of Mormon. The first is that it appears in the original text (in some form or other - who knows what it would appear as in 'reformed Egyptian'). And in this case, the translation is a very literal rendering of the concept of synagogue into English. And the anachronism should be attributed to Nephi (and the other authors of the original text). Our second option is that the concept of synagogue (and everything that goes with it) isn't in the original text. There may be something similar (in terms of meaning), but the English translator chooses the word synagogue because it is understood to the intended audience. The anachronism occurs because of an English word choice, and doesn't necessarily exist as an anachronism in the original text. We make efforts to try and differentiate between the two choices (there is no third option really if we assume an ancient source).
So as an example. One of the features that is standard to a synagogue is an ark in which the torah scrolls are kept. If the Book of Mormon text were to talk about how the priest went into the Synagogue, went to the ark, and removed the scrolls to read to the community, it would be much, much more difficult to assert that the use of the word 'synagogue' on the part of the translator was simply a preferential selection of one possible word out of several similar choices. In this case, we would really view the term synagogue as a more literal translation of an underlying text. But, in my opinion, we don't have this sort of interaction in the text of the Book of Mormon for the word synagogue (in the twenty-five times that it occurs). We have a much stronger interaction of this sort in the term 'church'. So, I tend to place 'church' in (2) and 'synagogue' in (1).
Having said that, there are a couple of features in the text that are worth discussing. Consider the chronologically earliest instance in 2 Nephi 26 -
Behold, hath he commanded any that they should depart out of the synagogues, or out of the houses of worship? Behold, I say unto you, Nay.
See the parenthetical? Is this a part of the original text, or a part of the translation? It seems to read: "Behold, hath he commanded any that they should depart out of the synagogues (or out of the houses of worship)? Behold, I say unto you, Nay.
It could be either, potentially. If I were Nephi, and I was inscribing a text on gold sheets, and I made a mistake and decided to change it, what do I do? Or it could simply be a parenthetical (Joseph Smith seems to like these sorts of explanatory asides as evidenced by later texts). 2 Nephi 26 is a fascinating text in its own right because of its intertextual nature (I have a significant discussion of it in my postmodernist paper on pp. 154-156).
The text that brings 'synagogue' closest to (2) is Alma 16:13
And Alma and Amulek went forth preaching repentance to the people in their temples, and in their sanctuaries, and also in their synagogues, which were built after the manner of the Jews.
But this one has an interpretation issue. If the subordinate clause "which were build after the manner of the Jews" applies only to synagogues, then it is a stronger argument for (2). If on the other hand, it applies to all three - that is the temples, sanctuaries and synagogues are all built after the manner of the Jews, it lessens the impact.
And finally, we have in Alma 31, the discussion of the Zoramite synagogue, which is clearly very different from any Jewish synagogue - particularly in verses 12-13:
Now, when they had come into the land, behold, to their astonishment they found that the Zoramites had built synagogues, and that they did gather themselves together on one day of the week, which day they did call the day of the Lord; and they did worship after a manner which Alma and his brethren had never beheld; For they had a place built up in the center of their synagogue, a place for standing, which was high above the head; and the top thereof would only admit one person.
Likewise in Alma 21:4:
And it came to pass that Aaron came to the city of Jerusalem, and first began to preach to the Amalekites. And he began to preach to them in their synagogues, for they had built synagogues after the order of the Nehors; for many of the Amalekites and the Amulonites were after the order of the Nehors.
So another potentially different synagogue (one not built after the manner of the Jews). And yet this is still a synagogue. So the Book of Mormon clearly has a view that a synagogue is a place of worship. But it doesn't really seem to go beyond that, and it could be argued that it is a term that seems to be used rather generically. On the other side of the coin, the word 'church' seems to be connected to a great deal more in the text.
So this is the way that I approach it. It may not be right. I'm certainly open to discussion.
Was that better? Or am I trying to answer something that you aren't really asking?