Thanks for confirming the situation.
In the first instance, I would suggest that you send copies of the pictures to the British Embassy in Hungary and ask them to reconsider their decision.
Asking for a reconsideration should not cost you money, since EEA family permits are supposed to be issued free of charge.
If that does not resolve your issue, you should contact Your Europe Advice so they can issue a legal assessment that you can then send to the British Embassy.
If this does not resolve the issue, Your Europe Advice can refer you to SOLVIT, the EU's problem solving network. The Hungarian centre will then make contact with the UK centre to try resolve the issue and get the family permit issued. Hopefully this should resolve the problem.
If this still does not resolve the issue, you should resubmit a new application to the British Embassy (which should be free of charge) and then appeal any refusal to issue a family permit to your wife. An appeal usually takes 6 months.
I can send you the links to Your Europe Advice if you wish.
I hope this answers your question.