Here is the item again:
Mrs. Neal loves chocolate cake. (A)
But couldn't accept the slice that her student Michael
offered (B)
because she knew that it was a bribe
(C) for a better grade
on his midterm essay.
You correctly realized that the period after cake creates a fragment. But couldn't accept the slice ... is a lonely verb phrase. When you attach a phrase of this type to the end of a main clause, you need no punctuation.
To fix the problem, you chose to do this:
Mrs. Neal loves chocolate cake but couldn't
accept the slice that her student Michael offered because she knew that
it was a bribe for a better grade on his midterm essay.
You have correctly employed Punctuation Rule 8:
Main clause + Ø + lonely verb phrase.
What an intelligent decision!