- The only way to put Briala truly in control is to let Celene die and set Gaspard up as a puppet. If you allow Celene to die, Gaspard will become emperor, but you'll get the option to either allow him to rule in truth, or to give Briala blackmail material and force Gaspard to be her puppet. Presumably, the latter requires that you actually found dirt on Gaspard, but that's not hard.
- Alternately, you can support Celene and reconcile with Briala. If you support Celene, you'll have the opportunity to EITHER reconcile her with Briala but allow Gaspard to be executed, OR you can save Gaspard's life by asking Celene to exile him. If you choose the latter, Briala will get nothing, so you can't both save Gaspard and support Briala if Celene remains in control.
- Or, it's possible to force all three of them to work together. That requires gathering blackmail material on all of them, so it's a little tricky.