I DO think it’s okay to say “A fool could fall and fall hard.” However I would’ve dressed it up a bit and said, if in a story, “A fool can fall and fall just as hard as anyone else.” Whoa… too long. “A fool can fall and fall more quickly.”

That seems to take in the sense of a fool falling, too… taken from that old saying ….something ‘hardest before the fall.’ You know what I mean, I hope. It’s about how even the wise or rich can fall, and …something…hardest before the fall. Oh well. Anyway, the fool would have a shorter distance to fall, presumably. I hope this bit of foolery helps.

The original sentence you questioned is fine.

no, i think it is right wouldn’t an equivalent version would simply be

A fool could fall and could fall hard

but if you try yours you get

A fool could fall and could falls hard

see yours doesn’t work