I have spoken verbally to multiple members of Apple's support teams. Apologies if that was unclear. I didn't record this calls, as they did not permit me to.

Then continue that. It’s definitely a fraud detection lockout due to probably the retailer not properly registering the gift cards or something similar.

If you can’t iforgot.apple.com, then support is your only option. No one else has access. Only Apple Support.

You have a case number, keep calling every 12 hours asking for an update.

You don't need their permission to record the call if they are recording the call already.