Is it obligatory for someone who performed tayammum without basmala due to ignorance of its ruling to make up for those prayers, and if so, how should they make up for them without knowing the exact number of times?

"Bismillāh" (mentioning God's name) when performing tayammum is a Sunnah, not a condition or an obligation according to the majority of jurists (Mālikīs, Shāfiʿīs, Ḥanafīs, and one narration from the Ḥanbalīs). Thus, tayammum is valid without it, and nothing is incumbent upon the one who performs it.