A few thoughts:

(1) יוֹם הַשִּׁשִּׁי (*yom hash-shish-shi*) is a construct phrase where the adjective שִׁשִּׁי acts as a noun. The phrase literally means "the day of the sixth" or "the sixth's day" but can simply be understood as "the sixth day". This kind of construct where the last word is an adjective is not ungrammatical, but it is relatively rare. Here are the only eleven verses that I found in a computer search where there is no definite-article/definite-preposition preceding *yom* + definite article + adjective (i.e., adjectiveיום ה): Gen 1:31; 2:3; Ex 12:15, 18; 20:10//Deut 5:14; Lev 19:6; 22:27; Lev 14:57; Josh 24:31; Judg 2:7. In the last three passages the adjective is not a number. For a few other similar cases see BDB 2b: https://archive.org/details/hebrewenglishlex00browuoft/page/208/mode/2up. See p. 209. 

(2) It is also relatively rare for there to be a definite-article + *yom* + definite-article + adjective (i.e., adjectiveהיום ה) though this kind of adjectival construction (i.e., adjectiveה nounה) is common overall. I only found four occurrences and they all (coincidentally?) have *rishon(im)* ("first)" or *acharon* ("last") for the adjective: Num 6:12; Eccl 7:10; Dan 10:12; Neh 8:18. Instead, the common pattern to express definiteness for *yom* in the HB is a definite-preposition + *yom* + definite-article + adjective. I found 117 verses like this. For example, בַּיּוֹם הַשְּׁבִיעִי ("on the seventh day") in Gen 2:2. There is probably nothing too significant about this since it would be more common to speak about a particular day using a preposition than not. Nevertheless, it is interesting to note that *adjectiveהיום ה* is even more rare than *adjectiveיום ה*. 

(3) I agree with the verse division in Gen 1:31–2:1 since 2:1 is a new thought. The narrator is now telling us that the creation is finished. 

Compare Gen 1:31–2:1 and Gen 2:3a:

Gen 1:31–2:1:

> וַיְהִי־עֶרֶב וַיְהִי־בֹקֶר **יוֹם הַשִּׁשִּׁי**׃
> 
> **וַיְכֻלּוּ הַשָּׁמַיִם** וְהָאָרֶץ וְכָל־צְבָאָם׃
> 
> And there was evening and there was morning, **the day of the sixth**.
> 
> **And they were finished the heavens** and the earth and all their army. 

Gen 2:3a:

> וַיְבָרֶךְ אֱלֹהִים אֶת־**יוֹם הַשְּׁבִיעִי וַיְקַדֵּשׁ אֹתוֹ**
> 
> And God blessed **the day of the seventh and he sanctified it**. 

Notice that both have the *adjectiveיום ה* construct phrase but only the first passage has the acrostic and it is broken into two thoughts. If the acrostic were intentional then it seems to me that it would be in a passage like 2:3a since it would be continuous with itself. It would also parallel *elohim* in the same verse. Because it is not, I'm inclined to think that the acrostic is not intentional. 

(4) Perhaps Nahum Sarna is on the right track to suggest that the definiteness of the construct phrase for days six and seven might be due to emphasize these culminating days of creation. I also suggest that perhaps style could explain the choice of the definite construct phrase rather than the definite adjectival phrase. Notice that seven of the eleven *adjectiveיום ה* verses cited above are considered to be from P: Gen 1:31; 2:3; Ex 12:15, 18; Lev 14:57; 19:6; 22:27. (Ex 20:20 // Deut 5:14 might also be from a priestly editorial hand.) And one of the four *adjectiveה nounה* verses is considered to be from P: Num 6:12. Thus at least for the word *yom* there could be a preference in P for the *adjectiveיום ה* construction where a definite preposition is not used. Just a thought.