The hadith literature was compiled two centuries after Muhammad. You can’t treat the hadith as unreliable when convenient and reliable when not.

Consistency matters here. The hadith is used because Sunni Islam itself treats Bukhari as the most authoritative source after the Qur’an, so the consistency question is internal to Sunni Islam and does not depend on a Christian accepting the hadith’s reliability. The framing is by the standards Islam itself applies to these sources. A Muslim who rejects the reliability of Bukhari has a different problem, since the Sunnah, and with it much of Islamic law, rests on the same literature.