The tricky part about with your conversion to reported speech is that the antecedents of both instances of "she" are ambiguous. Splitting up the reported speech into two sentences accentuates the ambiguity.
Ms. A said that she had given up on the justice system and on the government. That's why she committed suicide.
It is no longer clear that Ms. A is talking about someone else. It's possible that Ms. A committed suicide, and you are reporting what she wrote on her suicide note. It's also possible that there are two people, and one committed suicide due to the other's loss of faith in justice and government, but that's less logical after some interpretation.
So, you have two problems to fix: one of clarifying the antecedent, as well as your original question about the sequence of events. You can address both problems by restructuring your sentence:
The victim committed suicide because she had given up on the justice system and the government, according to Ms. A.
Using the word "victim" makes it clear that Ms. A is not the one who committed suicide. Placing the "Ms. A" at the end of the sentence also helps to link the pronoun "she" with "victim" instead of with "Ms. A", although the placement at the end is optional. Finally, to address your original question, Ms. A is reporting a cause-and-effect relationship, so you can use "because" when paraphrasing her speech.