A major challenge when studying the Rambam's Mishneh Torah is figuring out his sources. Countless commentaries and volumes have been written on finding the Rambam's sources. A contemporary example is Rabbi Chaim Kanievsky's קרית מלך. It's a small volume, and very concise, so it takes some work to figure why he considers a particular text the source for a halacha in the Mishneh Torah. In that volume he points to the end of the Midrash Tanchuma in Toldot and comments "צריך עיון" which indicates that even that source is not so clear.
That Midrash lists the virtues of the Messiah:
ונשא ממשה שאמר (במדבר יא) כי תאמר אלי שאהו בחיקך
"[And he'll be] loftier than Moses, as it's written (Num. 11:12) "[Did I conceive this entire people? Did I give birth to them,] that You say to me, 'Carry them in your bosom [as the nurse carries the suckling,' to the Land You promised their forefathers?]"
(It seems to me, that the Midrash is indicating that the Messiah is greater that Moses in that he will bring Israel into the Promised Land- something that Moses did not do.)
But it's not clear what this verse has to do with prophecy. (And that's probably why R' Kanievsky writes "צריך עיון"- the Midrash contrasts Moses' greatness with the Messiah, but not necessarily in regards to prophecy.)
Indeed this very question is dealt with by R' Tzvi Hirsch Chajes in his אגרת בקורת. Here too he struggles to find a source. He also mentions the above cited Midrash as a possibility, but rejects it for the reason I mentioned above: the Midrash is referring to some virtue that the Messiah has above Moses- but it can't be prophecy for we know that וְלֹא קָם נָבִיא עוֹד .בְּיִשְׂרָאֵל כְּמֹשֶׁה (Indeed the Rambam repeats this principle in his commentary on Tractate Sanhedrin and in Hilchot Yesodei HaTorah.)
The מהרי"ץ חיות then states that he's unaware of any source that states that the Messiah will be a prophet, rather he will be Divinely inspired as stated in Isaiah (cited in the question) and other verses:
באמת לא אדע המקום אשר מפורש בו שמלך המשיח יהי׳ גם כן נביא, רק שרוח
הקודש תנוח עליו ויתיחסו על פיו,
וזה מבואר בפסוק ישעיה ו׳, ונחה עליו רוח ד', ובפסוקים אחרים
The מהרי"ץ חיות then cites the Rambam in the Guide For the Perplexed:
The second degree [of prophecy] is this: A person feels as if
something came upon him, and as if he had received a new power that
encourages him to speak. He treats of science, or composes hymns,
exhorts his fellow-men, discusses political and theological problems;
all this he does while awake, and in the full possession of his
senses. Such a person is said to speak by the holy spirit. David
composed the Psalms, and Solomon the Book of Proverbs, Ecclesiastes,
and the Song of Solomon by this spirit; also Daniel, Job, Chronicles,
and the rest of the Hagiographa were written in this holy spirit;
therefore they are called ketubim (Writings, or Written), i.e.,
written by men inspired by the holy spirit.
ואם יקרא בקצת העתים נביא הוא לקצת כללות להיותו קרוב לנביאים מאוד אשר
מפני זה גם סדר דניאל וכתובים היו רק בעלי רוח הקודש אבל בכל זאת נקרא
נביאים בכלל יש לומר באמת כי משיח לא יהיה נביא ומש״כ הרמב״ם נביא פי׳
שרוח הקודש יהיה עליו ובמעלה הזאת יהיה קרוב למשה אבל לא נביא ממש
Sometimes, these individuals (David, Solomon & Daniel) are called prophets because that have achieved a level that is very close to real prophecy. Perhaps, R' Chajes concludes, Moshiach need not be a prophet, but must reach a level close to prophecy through his attainment of Divine inspiration- and thus approaches the level of Moses, while still not technically being a prophet. (i.e. when the Rambam writes "prophet" in the Laws of Repentance, he's using the term loosely.)
Return to your original question, it would appear that the verse in Isaiah could not necessarily prove that the Messiah will be a prophet, rather it shows that he will attain the lesser "prophecy" of Divine inspiration.