Take another look at the original sentence:
You wanted to fix it this way:
To delete the comma between puppy and she would cause an equally bad problem, a fused sentence. A fused sentence occurs when you join two main clauses with no punctuation whatsoever. Sherian refuses to let her kids have a puppy is the first main clause. She is convinced is the second main clause. The spot between puppy and she needs a stronger break than blank space.
You might want to consult the rules for fixing comma splices and fused sentences.