Your question comes from two points, that:
(a) Halachos L'Moshe MiSinai are not hinted to in the Torah
(b) that these halachos are therefore outside of and external to the Torah, and the completeness of the Torah would not allow for such a thing.
In terms of the first point, while that is clearly the opinion of the Rambam in his Introduction to the Mishna (8:22):
ועל כן כל דבר שאין לו רמז במקרא ואינו נקשר בו ואי אפשר להוציאו בדרך מדרכי הסברא עליו לבדו נאמר "הלכה למשה מסיני"
There are those who question the Rambam' assertion. For example, in אור זרוע לצדיק R' Tzaddok writes:
עוד כתב רבינו שם דהלכה למשה מסיני אין לו רמז בקרא, וגם על זה אני תמה מהא דסוכה ל"ד... ועוד קשה לי מהירושלמי
The Rambam writes further that _Halacha L'Moshe MiSinai has no hint in Scripture, and also on this I am surprised, for we find in Sukka ... and in the Yerushalmi ...
Regarding the second point, I think a comment of the Alshich on Devarim 17:11 might help address the question, regarding the injunction to listen to the Sages:
אשר יגידו לך - הלכה למשה מסיני, וגם אם לא יהיה לה טעם בעיניך, וכאומרים על שמאל שהוא ימין, כי במה שאומרים בהיקש או על פי מדות שהתורה נדרשת וגזירות וסייגים בודאי נראה הטעם
"That which they tell you" - Halacha L'Moshe MiSinai, and even if you don't see any reason for it, and when they tell you about left that it is right. For, in that which they tell you through a juxtaposition, or through one of the methods of expounding the Torah, and fences, certainly one sees a reason.
What I understand the Alshich to be saying is, that Halacha L'Moshe MiSinai is a very important part of the Torah itself, namely in that it demonstrates that there are some things which are to be accepted simply because they are coming through Rabbinic tradition, and Rabbinic tradition is an essential facet of the Torah. Halachos L'Moshe MiSinai, not in content but in concept, are included in the Torah, in the injunction of אשר יגידו לך. An element of the completeness of the Torah is the inclusion of the Mesora on which it relies.