Upvote:6
Contact the Hungarian Embassy in Washington DC
The ONLY solution to this is to gather facts regarding border entry and governance of borders. Contacting the Hungarian Embassy will allow you to share your story and the difficulty, you can sprinkle in some tears if you think it will be effective.
You will want to find out
Length of time to return
Audit trails they require
Secondary methods
Temporary visa's
If that doesn't work, you can call any European Embassy to gather more information and see if there is some assistance you can get from them. Hungary needs USD and my suspicion is that you will get assistance and this will be a matter for the dust bin poste haste.
Do it correctly. Reduce Risk.