Appealing is useless. She holds a British passport and, therefore, should be rejected. I used to be in the same situation as her. Now, I only have Irish citizenship. Please note that what your wife will need to do, IF YOU CHOOSE THIS ROUTE, is to renounce her British citizenship. It costs 187 pounds and my application took 2.5 months to resolve. There is more info here:
https://www.gov.uk/renounce-british-nat ... y/overview. PLEASE NOTE: it's not easy to get British citizenship back if she changes her mind in the future. In fact, there doesn't seem to be any way to get it back if you only renounce because you don't want to be British any more (as opposed to if you renounced it in order to take on a new citizenship of a country which doesn't allow dual citizenship).
Apparently, there have been many people who have already renounced their British citizenship and their family members have subsequently successfully applied for the EEA family permit. I haven't reached this stage yet.
There is a Facebook group about EEA applications and a few people have mentioned renunciation of British Citizenship there. You might want to search on there. I don't have the address so you will have to look for it by yourself or wait for someone else to post it here.
Don't forget there is another option open to dual nationals, i.e. the Surinder Singh route, wherein the dual national moves to another EEA country (not one of which they are a citizen!) and exercises treaty rights there (i.e. working/self-employed/studying with health insurance/self-sufficient with health insurance). The issue with this is that no-one knows for sure how long you would need to stay in the other country. Also, the non-EEA family member must live with them in the other country. They also need to prove that they have moved their life to the new country, e.g. by learning the language/joining clubs/having bank accounts/applying for residence documents. This won't suit everyone, especially if you have kids.
Hope this gives you a few ideas. Consider carefully!
Good luck!