The way that you translated it might seem to indicate that R. Chanina was reading the Beraita. However, we can challenge your translation. You appear to have rendered the word קרא as a verb meaning that R. Chanina was reading. However, we know from elsewhere in the Talmud that the word קרא is actually part of R. Chanina's name/title. In Taanit 27b we find:
והאמר רבי חנינא קרא צער גדול היה לי אצל רבי חנינא הגדול ולא התיר לי לפסוק אלא לתינוקות של בית רבן
In that instance it seems pretty clear that קרא is not a verb, but is instead a name or an appellation for R. Chanina. Indeed, Rashi tells us as much in his commentary there:
שהיה בעל מקרא ויודעה בגירסא ובקי בטעמיה
So getting back to the passage in Yevamot, the opening phrase would just mean that R. Chanina Kara was sitting in front of R. Yannai and saying that the halacha follows R. Yehuda. Indeed, this is how Soncino translates it (with the appellation קרא translated as well):
The Bible teacher, R. Hanina, once sat before R. Jannai, and as he sat there he stated: The halachah is in agreement with R. Judah.
This is also how it is translated in the link to Sefaria in your post:
Rabbi Ḥanina Kara, the Bible expert, was sitting before Rabbi Yannai, and he was sitting and saying: The halakha is in accordance with the opinion of Rabbi Yehuda.
As for R. Yannai's response, which you translated as "go out to read your reading", which could theoretically have referred to a Beraita if that's what he was reading, the two aforementioned translations both interpret that as referring to R. Chanina's reading of verses. Soncino has it as:
The other called out to him: Go out, read Biblical verses outside; the halachah is not in agreement with R. Judah.
Sefaria has it as:
Rabbi Yannai said to him: Leave the study hall and recite your verses outside, as you are incorrect in your ruling; in fact, the halakha is not in accordance with the opinion of Rabbi Yehuda.
These translations make sense, because since we rendered קרא as a noun, there is no indication that R. Chanina was reading anything about R. Yehuda. All we know is that he stated that the halacha follows R. Yehuda. In fact, in the Soncino translation to a parallel passage in Ketubot 56a, the footnote gives an explanation for why R. Yannai would specifically send R. Chanina outside:
[I.e., Go teach the Bible to children instead of venturing into the realms of the halachah. Bible instructions were given in a place 'outside' the academy].
In other words, R. Yannai was specifically telling him to read Scripture outside, and it had nothing to do with reading anything about R. Yehuda.
If this is the case then there would be no indication from this passage that R. Chanina was reading a Beraita.
