Mustang knew the Bellona would try to make a move against House Augustus but would need Lune backing to do so. She tried to ingratiate herself with both House Lune and House Bellona (via Cassius) to either prevent said action or at least catch wind of what was going on before hand and alert her father. Her plan obviously backfired as Octavia went behind her back negotiating with Bellona and (presumably) the Bellona kept Cassius in the dark to keep it away from Mustang.

It's not a terrible plan in theory but Mustang was probably a little naive to think she could ingratiate herself with either Lune or Bellona quickly enough to be "in" on any plan against her family. Perhaps she thought she'd have years to build trust with both parties before it happened in which case it probably would have worked.