IIRC the ignition on the car is not fused. the fuel pump could be the culprit but i doubt playing with a fuse would change it.

and if there's enough juice in the battery to turn the engine over, there's definitely enough juice to give it spark. turning the engine over requires lots of current.

although trouble shooting modern engines is a lot more difficult, you need to start with the basics. when the problem occurs, do you have spark? do you have gas (getting to the cylinders)? very often, one of those two are missing.