Thanks to the "heads-up" I got from your comment on my answer to your last question earlier, I had a chance to do some research on this today (although I'm not sure about your page numbering in Breasted. You may have a different edition to mine).
While translations like the ones in Breasted (pp 175-192) and Lichtheim (pp 29-35) are excellent resources for the study of the Battle of Megiddo (not to mention the study of ancient Egypt in general), they can sometimes create the impression that we know far more detail than is actually the case.
Of course, many modern websites are even worse in this regard!
Details in the original texts are often difficult to interpret. There can be a number of reasons for this. Firstly, descriptions of the battle and subsequent siege are much less complete than we would like. They frequently give us just the highlights where we would prefer to know more detail.
For example, we are told that, during the siege, the Egyptians constructed a timber wall and ditch surrounding Megiddo. The trees used to build the wall are described as "bnr" ("sweet") which, in this context, could mean that they were fruit trees. If so, the Thutmose's troops probably destroyed the orchards that surrounded Megiddo to construct the wall, adding further to the city's woes. However, we cannot know for certain because the texts are just not explicit.
We also know that the siege lasted seven months. It seems reasonable that the Egyptians would also have built other structures as part of their siege-works (perhaps defensive "camps" or barracks for their own troops, for example), but if they did, the texts tell us nothing about them. This point is particularly relevant to your question.
The texts themselves are usually damaged. Lacunae (holes) in manuscript texts mean that words, or even entire sentences, are often missing. Inscriptions on temple walls are frequently damaged. Where this damage is relatively recent (as with the inscriptions at Karnak), we can sometimes make use of copies made by early Egyptologists. In this instance, we are fortunate to have a copy published in 1906 by the German scholar Kurt Heinrich Sethe in his Urkunden der 18 Dynastie, which is generally considered to be an accurate record of the inscription at Karnak as it survived at that time.
While double-checking some details online, I found this web page by Rhio H. Barnhart which offers some details on the process of transliterating and translating the Karnak inscription of the events of the Battle and Siege of Megiddo. [I have a great deal of sympathy, since I had to do something very similar for one of my assessments when I studied Egyptology!]
So what about this "fortress"?
We know that the "xtm" (probably a "fort" rather than a "fortress") in which Thutmose stayed during the Siege of Megiddo was on the east ("iAbty") of the town. That much is made explicit in the inscription:
His majesty, himself, was on the fort east of this town. He was
watching over it, night and day.
There is nothing to say whether the fort was one captured before the siege, or built by the Egyptians themselves as part of their siege-works. Personally, I suspect the latter, but once again, the texts are not explicit.
In your comment to my last answer, you mentioned a thick wall described in the texts. In fact, there is an entire line missing between the mention of the fort quoted above and the description of the wall. That description is itself damaged, but it seems that the Egyptians named the wall "mn-xpr-ra iH sTtyw tA" ("Menkheperre-enclosing-the Asiatics" - Menkheperre being Thutmose III's prenomen). If this is correct, (as I say, the text is damaged, but this is how I would read it) it seems more likely in this context that the "thick wall" was the one surrounding Megiddo, rather than a reference to the walls of the fort.
Now, I know that some historians have argued that Thutmose stayed in Jerusalem during the siege. I think this may be based on a misreading of the text (and, possibly, some confusion about geography).
Middle Egyptian has several words for "fortress", including "xnrt", "itH", and "mnnw", depending on context (these are transliterations. a Google search will probably give you the hieroglyphs for each word). In this case, the text is explicit. As I mentioned above, the word used is "xtm", which is only used elsewhere for a "fort" in a military context.
Other towns are explicitly named in the texts. If Thutmose had stayed in Jerusalem, why wouldn't the scribes have simply said so? Why would they just describe it as a "fort"? Besides, Jerusalem is far to the south of Megiddo. The text states explicitly that the fort was to the east.
Furthermore, we know from this and other sources that Thutmose III had cultivated his reputation as a warrior. It seems likely he would want to remain close to his army while on campaign outside Egypt.
Now, as I've said several times, we can never be certain. The texts are just not sufficiently explicit. But with that caveat, it seems to me far more likely that he remained with his army, in a "fort" constructed as part of the Egyptian siege-works, and located to the east of Megiddo..
Breasted, James Henry: Ancient Records of Egypt Volume 2: The Eighteenth Dynasty, University of Illinois, 1906
Lichtheim, Miriam: Ancient Egyptian Literature: Volume II: The New Kingdom, University of California, 1976