During heavy rain, my basement floods with about three inches of clean, non-smelly water up through the floor drain, which is about six inches square. I have stood and watched this water come up. I had the drain snaked and the auger came back up with mud (broken trap?). Can I solve this problem by digging up the trap and replacing with a sump pump?

Sounds more like you floor drain drains to day light. Is that possible, how far did you snake?