When Vatican City became an independent state in the 1920s, how were its boundaries determined? Why was the Castel Sant Angelo excluded?

The Lateran Concordat or Lateran Accords were signed on February 11, 1929 by Benito Mussolini representing the Italian State and Pietro Gasparri representing the newly-created Vatican state. Article 3 of the treaty states:

Italy recognizes the full ownership, exclusive dominion, and sovereign authority and jurisdiction of the Holy See over the Vatican as at present constituted, together with all its appurtenances and endowments, thus creating the Vatican City, for the special purposes and under the conditions hereinafter referred to.

The boundaries of the said City are set forth in the map called Annex I of the present Treaty, of which it is forms an integral part.

It is furthermore agreed that, although forming part of the Vatican City, St. Peter's Square shall continue to be normally open to the public and shall be subject to supervision by the Italian police authorities, which powers shall cease to operate at the foot of the steps leading to the Basilica, although the latter shall continue to be used for public worship. The said authorities shall, therefore, abstain from mounting the steps and entering the said Basilica, unless and except they are requested to do so by the proper authorities.

Should the Holy See consider it necessary, for the purpose of special ceremonies, temporarily to prohibit the public from free access to St. Peter's Square, the Italian authorities shall (unless specially requested to do otherwise) withdraw to beyond the outer lines of Bernini's Colonnade and the extension thereof.

As you can see, there was a map of the territories that now constituted the Vatican. Wikipedia provides us a half decent version. The negotiations for just where those borders would lie took a good deal of time, and I have not studied the negotiations themselves. But, from the document we can glean some concerns.

The Italian state wanted to keep access to St. Peter's Square, and put that in writing in the aforementioned Article 3. So, you can see on the map that it is the very border of the Vatican. Now, Castel Sant'Angelo is "down" from St. Peter's Square on the map I previously linked, and is to the east (right) on most maps. So, to extend the territory of the Vatican to Sant'Angelo you would have a bit of a hard time designing a plan that would not impede Italian access to St. Peter's Square. The addition of that location would also be a fairly large increase in the size of the Vatican state--if you look at maps you can see that Sant'Angelo is somewhat more than 25% or so of the Vatican's territory in Rome (to do a very rough visual estimate).

Why exactly the Vatican didn't fight to get it included in their territory, I do not know. Given its history as a mausoleum and fortress/castle and current use as a museum, I would only feel confident saying that the defensive utility of the location had long since been unimportant (at least going back to 1870). The papacy had by 1929 given up the idea of defending itself by force.

I am certain that there has been some scholarship done on the negotiations themselves, given their endurance to this day and the colorful figures involved. This particular bit of history has only been of tangential interest to me (Article 24 in particular) so I haven't read further on the minutiae of the negotiations as of yet. I hope I have answered your question at least in part, and as always followup questions by OP and others are encouraged!

(edited for formatting)