I don't want to be negative but as you have to request this EEA family permit to UK administration, they will surely discover you're holding also a UK citizenship and passport (so you may be denied). EEA Family Permit is only for citizen from an EU country living in another EU country.
Other disadvantage (apart the fees) for the EAA family permit is it's valid for 6 months period and have to be renewed every 6 months.
And last point, it will allow your partner to stay in North Ireland (and travel in all UK) but not to the rest of Ireland. For this a Visa from Ireland will be required.