This is a tough one. The biggest problem is that the fuse doesn't pop until you load the electrical system. Could be anything from the items you mentioned to the charging system, chassis grounds, integration relay to the ECM itself or more.
Anything that happened to the car immediately before or after this started to happen? Any aftermarket electronics wired to the car, jumped the car recently, new battery, repairs, etc.?
If the fuse popped right away - that would shorten down the list of possible culprits. This "delayed" reaction to popping only under a load makes it tough to nail down.
You can run the car for a short time with the serpentine belt off the car - that should tell you if the alternator is suspect. Coils and pump would be a bit tougher - hard to test without swapping in a known good one - but if I had to guess, I'd pick the fuel pump as the most likely out of those two. Can try disconnecting the pump to see if it pops the fuse or not - you can plumb in one of those fuel injector service kits to run the engine for a short time. That should verify if the pump is seizing up on you or not.
From how this sounds - could be the ECM. The 9th gen Corollas had more problems with bad chips / cracked board in the ECM popping fuses than the 8th gen - but checking on the wiring harness running to the ECM couldn't hurt. Also check the wiring running to the wiring block on the under the driver's side dash - look for any discolored or burnt wiring.
A Toyota handscanner can query the ECM to see if it is working correctly, but to get that checked you have to bring the car to the dealership.