Honestly, I haven't seen that many Flippy-centric fics that revolve around conflict in the traditional sense? Maybe I'm just looking in the wrong corners. A lot of the plots I stumble into are more about the aftermath and the psychological toll—Flippy trying to reintegrate, or Flaky quietly worrying about him, or him having a near-breakout moment that she manages to soothe. The conflict is almost entirely internal, the struggle against the trigger and the guilt. Sometimes another character will poke the bear, intentionally or not, and that's where the external tension comes from, but it's usually a catalyst for exploring his instability, not a prolonged rivalry.
I did read one where a new, aggressively chaotic character kept deliberately setting off Fippy to 'test' him, which created this exhausting cycle of violence and recovery. It was less about them fighting and more about the emotional drain on everyone trying to protect the new guy from himself and from Flippy. The conflict was the unsustainable situation, not a battle to be won. Most writers seem more interested in the quiet, fragile peace between his two states than in manufacturing big showdowns.