In his book "The Temple" Joshua Berman deals with this question around page 13-17.
Shabbat and the Temple/Mishkan are directly interconnected with each other. While we do things which are not allowed on Shabbat outside of the Temple, the construction of the Temple itself can not be done on Shabbat. One of the purposes of Shabbat is the Temple, and one of the purposes of the Temple is Shabbat. (One name for the Temple is the Beit Menucha)
By violating Shabbat for the activities of the Mikdash , and by defining Shabbat by the activities of the Mikdash, this connection and purpose can be felt directly.
And what is the connection between Shabbat and the Temple that is so important? Both are the pinnacle of the world's creation, and the defining thing which teaches us about the concept of Kodesh.
Shabbat is the first thing that is called Kodesh in the Torah. The Temple is the only thing called Kodesh Kodeshim. It is argued, that the completion of the creation of the world was not completed until the Mishkan was built. This is reflected in the psukim:
Bereshit 1-2 Shemot 39-40
1:31 And Gd saw all that he had made and it was very good 39:43 Moses saw all of the skilled work
2:1 The Heavens and the Earth and all of their array was completed 39:32 All the work of the Mishkan and the Ohel moed was completed
2:2 And Gd completed all of the work 40:33 And Moshe completed the work.
2:3 And Gd blessed 40:33 And Moshe Blessed
2:3 And sanctified it. 40:9 and you shall sanctify it.
This is continued in the book of Kings, where the number 7 is used for the construction of the Temple. It took 7 years to complete, It was dedicated on Sukkoth the 7 day holiday on the 7th month of the year, and the dedication is composed of 7 petitions.
The idea that the Temple is the completion of the creation of the world, is also explicit in a Midrash on the Book of Kings: Pesiktah Rabbatai 6.
"All the work the Sholomo did was complete. The Midrash says that it
does not say All work, but rather all THE work, just as it says in
Bereshit regarding Shabbat and the creation of the world. When Shlomo
finished the construction of the Temple, Gd said, "Now the work of the
heavens and earth are complete (shlema), and this is why Shlomo, was
tasked with it's construction. (Shlomo comes from the same root word.)
Another contrast. While Gd alone created the world, only man can build the Temple. When a perfect and just society is built, with the Temple at its center, then the work of the world is complete.
R. Ovadiah Seforno, says that the word Menucha does not just mean stopping vigorous activity, but also means that you feel a sense of completion. It is for this reason that when King David moved the Ark to Jerusalem, he called out in Psalms, 132:8, that he has brought Gd to his resting place (li-mnuchatekha) In Chronicles 2 6:41, Shlomo says the same thing, "Advance, O Lord Gd, to your resting place", Shabbat is also called a day of Menucha. Since they serve the same purpose, you must violate Shabbat to bring a Korbon, in the place of rest, and to bring Menucha to your soul.
And lastly, When Betzalel builds the Mishkan, it talks about his Melakhah, and so too on Shabbat, we are told to stop doing Melakhah. No Melkhah can be done to create the Temple on Shabbat, but on Shabbat you may do Melkah in the Temple. As a reminder of the Brit between Gd and the Jewish people, towards the creation of the world and a perfect society.