Agree with all the posters who ask why the OP wants to be able to describe herself as 'married' yet not do it legally.
Got married myself recently and the true nature of what it means is made quite clear - you contract a marriage and, totally aside from the romantic thing of being in a committed relationship (which I'm sure is the OP's situation), the marriage contract changes your legal status with regard to a variety of things - i.e. next of kin, inheritance, rights surrounding any children, all of which is an official recognition of the permanence you both intend to have about being together.
I didn't need a piece of paper to prove my emotional commitment to my partner and family, but we thought that it was better for all involved to change our status in a way we thought would better protect eachother and the children.
The GROIreland website explains all the new rules about the event called marriage in this country and, knowing the position of most churches in these things, I doubt they would entertain someone trying to describe what is essentially an unofficial commitment as a 'marriage'.
The OP probably has her own reasons why she doesn't want to change her legal status and it sounds like she wants to organise a religious commitment ceremony. Good luck finding a priest to do that, but there may be members of more obscure churches or individuals who would.