Do “because so-and-so claims that” and “claiming that” have the same meaning?

Your two sentences could have the same meaning, but it depends on how the grammar of the second sentence is interpreted.

In your first sentence, because she claims that her personal information was leaked is an adjunct of reason, stating why she filed a complaint to the committee.

In your sencond reason, claiming that her personal information was leaked is a gerundial non-finite clause which can be interpreted in 3 ways as:

1. attributive, modifying the noun complaint, stating what kind of complaint she filed.
2. an adjunct of manner, stating in what way she filed the complaint.
3. an adjunct of reason, stating why she filed a complaint to the committee.

If you chose the 1st interpretation you can understand:

She filed a complaint to the committee which claims / in which she claims that her personal information was leaked.

If you chose the 2nd interpretation, then the sentence has the meaning:

She filed a complaint to the committee (by) claiming that her personal information was leaked.

Finally, if you chose the 3rd interpretation then yes, the two sentences are synonymous, and you can move claiming that her personal information was leaked at the beginning of the sentence:

Claiming (because she claims...) that her personal information was leaked, she filed a complaint to the committee.