Is the oath sworn by the woman on the Mushaf that her husband's sister "did not tell me: I am the one who invited so-and-so" valid, given her knowledge that her husband's sister was the one who wanted the invitation, and is that considered a sin requiring repentance?

If the oath on the Quran was to prove a right, and you broke it, then this is a ghāmus oath (a false oath taken knowingly), which necessitates repentance to Allah, because an oath is based on the intention of the one who administers it. It is more prudent to offer expiation for an oath.