"We Are Dealing With A Student Standing Before His Teacher"
How are to resolve this difficulty of conflicting Tannaitic sources? On one side of the divide, the brayta states that one must not stop learning to physically destroy chametz but should rather nullify it. On the other side of the divide, the Tosefta clearly states that one should stop learning in order to physically destroy chametz, and derives from this the first law of the Mishna, which leads one to conclude that the Mishna is in agreement that one should stop learning in order to physically destroy chametz.
There are typically three paths to resolving apparent Tannaitic contradictions. One can claim that the sources are dealing with different cases. One can claim that this is in fact a matter of Tannaitic dispute. Or, one can emend the text of one of the sources, claiming that it was reported wrong.
I would posit that Rav Yaakov bar Acha was troubled by the apparent conflict between Tannaitic sources which we have discovered. The brayta argues with a stam mishna as well as with a Tosefta! His answer is that הכא בתלמיד יושב לפני רבו עסקינן. That is, the case in the brayta which states that one should not leave the study hall but should rather simply nullify is dealing with a student sitting in front of his teacher. There is the honor due his teacher as well as the opportunity to gain important Torah knowledge he might otherwise not gain. Staying before his teacher is so critical that it is just like the case of להציל מן הנכרים ומן הנהר ומן הלסטים ומן הדליקה ומן המפולת! Therefore, he should not leave the study hall but rather he should immediately nullify it in his heart.
Indeed, as the gemara notes, this can be derived from close analysis of the brayta.
דיקא נמי דקתני היה יושב בתוך בית המדרש שמע מינהAfter all, the brayta begins היה יושב בבית המדרש (footnote: or, according to the gemara's citation, היה יושב בתוך בית המדרש) and the word יושב can have a technical meaning of someone "sitting" before his teacher. Indeed, in the Tosefta we cited above, we have Rabbi Eleazar bar Tzadok and his father sitting before Rabban Gamliel in the study hall, thus demonstrating this usage: