The Brit Chadasha does not say that Yahushua KEPT the feast of Hanukkah.
Lets look at the KJV to see where all the confusion is.
22And it was at Jerusalem the feast of the dedication, and it was winter.
23And Jesus walked in the temple in Solomon's porch.
You can see that this text does not say that Yahushua KEPT anything, but simply says that he "walked in the temple in Solomon's porch."
Also, do some research and you will find that the words "the feast of" do not actually appear in the Greek text from which this verse is translated.
Here is an actual word for word translation of the Greek text:
And occurred the dedication in Jerusalem and it was winter.
And walked Yahushua in the temple in the porch of Solomon.
People assume that this dedication is the Feast of Hanukkah, but as the text indicates this was not a feast.
The Tanakh lists several dedications, none of which are the Feast of Hanukkah.
See Numbers 7:84-88, 2 Chronicles 7:9, Ezra 6:16-17 and Nehemiah 12:27.
We have 2 clues to help us discover which dedication is being referred to in John 10:22-23.
The first clue is that the text tells us that it was WINTER.
The second clue is in the next chapter in John 11:55 when it tells us that "PASSOVER was nigh at hand."
With these 2 clues we can now narrow this DEDICATION down to the one referred to in Ezra Chapter 6 when the Second Temple was dedicated:
15And this house was finished on the third day of the month Adar (WINTER), which was in the sixth year of the reign of Darius the king.
16And the children of Israel, the priests, and the Levites, and the rest of the children of the captivity, kept the DEDICATION of this house of God with joy.
17And offered at the dedication of this house of God an hundred bullocks, two hundred rams, four hundred lambs; and for a sin offering for all Israel, twelve he goats, according to the number of the tribes of Israel.
18And they set the priests in their divisions, and the Levites in their courses, for the service of God, which is at Jerusalem; as it is written in the book of Moses.
19And the children of the captivity kept the PASSOVER upon the fourteenth day of the first month.
20For the priests and the Levites were purified together, all of them were pure, and killed the passover for all the children of the captivity, and for their brethren the priests, and for themselves.
21And the children of Israel, which were come again out of captivity, and all such as had separated themselves unto them from the filthiness of the heathen of the land, to seek the LORD God of Israel, did eat,
22And kept the feast of unleavened bread seven days with joy: for the LORD had made them joyful, and turned the heart of the king of Assyria unto them, to strengthen their hands in the work of the house of God, the God of Israel.