Is it permissible for someone who went on Hajj at the expense of the state to consider her Hajj an obligatory one, or is that not permissible because she did not perform Hajj with her own private funds?

This Hajj is sufficient for the obligatory Hajj, as it is not a condition for the validity of Hajj that it be from the pilgrim's own money. However, spending from one's own money for Hajj is better and carries a greater reward with Allah, and what is important is that the earnings be good (halal (definition)).