I hope I'm not the bearer of bad news Michal but if you're a dual national then you can no longer rely on EEA rules
That means if you are both British and Irish, you can no longer use your Irish nationality for the EEA rules.
If you want to use the EEA rules and you hold an Irish passport, then you would have to revoke your British citizenship. Once you have done this you can't get it back, so think VERY carefully before you do it.
Alternatively, if you meet the new rules (which include a minimum income requirement of £18,600) then you can apply for her to join you under the UK laws.