In keeping with the plain meaning of the text the way Rashi says his commentary does, and in keeping with the Bereshit Rabbah 12:13 quoted, doesn't it seem more likely that he is saying that Avram concealed his wife, Sarai in "a word"? (Like in ראשי תבות, ראשי תיבות)
That if her status as his wife was revealed, the Torah would have said, "they came to Egypt..."emphasizing that they plural are together as a married couple and came to Egypt.
But by his concealing that status through the term he used to describe her (my sister), the Torah only emphasizes Avram's coming to Egypt individually (not as a married man) even though she accompanied him. When the tax collectors investigated, they discovered her true status demanding more taxes (שתבעו את המכס).
In other words, this story is discussing the idea of how taxes are determined (in both cases, truthfully) according to how you declare objects, something common to anyone who travels abroad or, for example, makes aliyah, as in immigration to another country.
If you declare a possession in a particular taxable category, for example a tapestry that can hang on the wall as artwork, the taxes will be higher. If declared in another category, in the example, it can also be declared as a rug to cover your floor, the tax is lower. In both cases, the declaration is truthful, but one will bring a much higher tax rate and the resulting financial loss.
In the case of immigration, taxes on property are often not applied for essential items that a single household brings with them. But duplicate items above those essentials are taxed. If Avram and Sarai immigrate as married, the duplicate items of Sarai's household would be charged tax to Avram. What's more, Avram was not entitled to use the items from Sarai's household to pay the tax charged to him because those items do not belong to him in the type of marriage they had.
Tax collectors try to determine if belongings are declared properly in order to collect more tax. Anyone who has been involved with aliyah knows this situation well. If declared wrong, it can cost a great deal of money needlessly. Like is known, causing needless financial loss is considered in halacha to be a danger to life (סכנה נפשות). See in this context Sefer S'dei Chemed, on the subject of selling a Beit Kenesset within a Jewish community where it results in a drop in the property value of those Jewish homeowners who remain behind.
Just to clarify further, early Indian (Ceylon) and Arab (pre-Islamic) practice at that time had 2 forms of marriage, Digah marriage and Binah marriage. In Binah marriage, the wife retains her own property, household and inheritance rights.
This is the origin of the term and idea that women, meaning wives are given "Binah Yiterah" like is found in Shas and elsewhere (See Niddah 45b, Sotah 35b and Torah Temimah to Devarim 11:19:6).
The wife in a Binah marriage is financially independent by law, more like a sister, than the wife of a Digah marriage, who is considered like property of her husband (her earnings and belongings become his).
It should be noted in this context, that the Matriarchs, Sarah, Rivkah, Leah and Rochel maintained their own households (their own tents) apart from the household of their husbands. In other words, the Avot were practicing Binah marriage as was the common practice where Avram and Sarai lived before coming to the land of Israel.
So in coming to Egypt, where the culture was different, Avram asked Sarai to declare herself as his sister, which was also true, in order that the entry taxes would be lower. They would be immigrating as two individuals with two separate households and the taxes charged to Avram would be substantially lower. Avram would not be killed (like mentioned above from the S'dei Chemed) with the additional taxes like is mentioned in Bereshit 12:12.
Additionally, both Avram and Sarai understood (they were both wise), before they ever reached Egypt, (see Ba'al HaTurim to Bereshit 12:13 to אחותי את) that it was likely that the Egyptians would try to acquire Sarai as a wife through the giving of gifts. She, at least, would be enriched materially. This is emphasized by both the commentary of Ba'al HaTurim to Bereshit 12:12 on the word יחיו and Rashi to Bereshit 12:13 on למען ייטב לי בעבורך. It wasn't a surprise. They planned together how best to approach this situation and to profit from it materially. This is what is meant when Avram emphasizes that his life would be bound/rolled up with Sarai's future (והיתה נפשי בגללך).
So to summarize, the translation of the Hebrew doesn't mean that Sarai was hidden in a crate by Avram, like some abusive husband (Even the expression from Bereshit Rabbah, "נְתָנָהּ בְּתֵבָה וְנָעַל בְּפָנֶיהָ" is only referring to applying the term my sister to her and covering her face with a veil which was the common practice for all modest women when in public.), but rather that she (meaning her status as Avram's wife) was concealed by a single word, saying that she was Avram's sister.