The fat for breakfast won't cause a painful flush, but it may cause an unsuccessful flush.
When fats/oils enter the stomach, a signal is sent to the liver to release bile. The reason to not eat any fats or oils the day of the flush is to build up the bile in the liver. The magnesium in the Epsom Salts allow the smooth tissue (bile ducts) to 'relax' so when the olive oil is introduced, the built up bile 'pushes' the 'stones' through the ducts. Relaxed ducts allow the larger 'stones' through easier.
Think of it as a water hose pushing a marble out. A trickle of water won't have the same push ability of a lot of water all at once.