You are doing it correctly: send your tax return in through the mail and claim your son. After your fiance's gets his full return back, he’s should amend his to totally remove the child.
Your return will NOT be denied or even delayed by the IRS. It will be processed in 6-8 weeks. You do NOT have to wait for your fiance's amendment to go through, or even be filed.
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If you and the other parent live together, only one of you can claim the child for any tax benefit. The TurboTax interview is confusing (it's designed for divorced parents). The second parent should not enter the child, at all. Doing so leads to mistakes. Your's is a common situation here, in this forum.
If you live together, you may decide between you who will claim the child. Only if you can't agree does the IRS have rules for who can claim the child. And those rules do NOT include "in WI the mother has sole and physical custody" .
1. The parent the child lived with the most, gets first claim. If that time is equal,
2. The parent with the highest income (AGI) gets to claim the child
