Given that your partner is Irish, and in view of the special arrangements between the Republic of Ireland and the UK, you do actually have a choice here.
That is you could either :-
- apply for an EEA Family Permit, or
- apply for a spouse visa under UK immigration legislation
There are advantages, and disadvantages. Firstly ... cost! The EEA Family Permit would be free, but the spouse visa would need to be paid for. However, secondly, if going the EEAFP route it would be 6 years before you might apply for British Citizenship, whereas going the spouse visa route, you might be able to apply for British Citizenship after only 3 years.
As they say ... you pays your money and takes your choice!
Again, in what way is your partner exercising EU Treaty Rights in the UK? Employed? Self-employed? Or how?