From NI but have Irish passport

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I am from Belfast but have only ever had an Irish passport and my declared nationality on the passport is Irish.

My wife is south African and we have never qualified to live in the UK as a married couple.

She has lived in the UK since 2013 but as a student. Her visa is expiring next year and she will have to leave.

I have recently been made aware of the ability for NI citizens to declare themselves as Irish and if necessary denounce British citizenship.

I am a student and do not meet the financial requirements to sponsor her as a spouse of a UK national.

Can she apply for an EEA residence card while already being in the UK on a Tier 4 student visa and claim to be the spouse of an Irish student living in England?

If not what other options are open to us that will allow her to not have to leave?

We only need her to have the right to remain until I finish my studies and then we will both leave the UK.

It's a shame as she only got her student visa so she could live with me while I studied as she was already a qualified engineering. The UK foreign office paid for her master's degree as a chevening scholar and then she is now finishing a fully funded PhD. It would've been cheaper for the UK if they had let her come over as my wife. But instead they paid for her further studies and now she'll be leaving the UK with those new skills.

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Can anyone here help with this?

Or can anyone please suggest a different area to post it in if this is not the correct section?

Thanks