We got a mortgage around the same time we got married and they talked us through what would happen if one of us died as a way to flog more life insurance to us. (At the time, we decided to wait until we have children before revisiting that kind of insurance) One of the things I had no idea about was that there is still a widow's pension equivalent which is about 200 euro/week which you get if you are married, but not if you are cohabiting. So if you died and you and your partner were married, then she would get 800/month until either she remarried or retired. If not, she would get nothing. Its not means tested, its based on your own PRSI payments. We lived together for years before we got married and I think if we'd known that we might have done it sooner. You can find the info if you google 'death related benefits'
Also, if one of you were to die unmarried, the survivor might really struggle accessing money as accounts would be frozen. With no will, money would not go to your partner, but I believe be held in trust for your children which wouldn't be helpful for her trying to feed them in the meantime. Finally, with your house, it makes a different if you're tenants in common or the other kind - I think that with one, ownership of the house would pass directly to the other partner, with the other they would be liable for inheritance tax and because you are unmarried the threshold is incredibly low.
In your situation, I would arrange a civil marriage and not tell anyone about it if I had ideological objections to marriage/didn't want to pay for a big wedding, etc. Arrange the party another time or forego it entirely, but you really haven't protected your children if you don't at least put alternatives in place.
Re: life insurance, it gets more expensive the older you get, but I think its a similar enough rate to the life insurance on the mortgage depending on level of protection - in the 30-50 euro/month bracket. I'd also consider insurance for what might happen if you were so seriously injured you couldn't work but didn't actually die. This unfortunately happened to someone I know - the mortgage insurance doesn't kick in, you can't earn but the mortgage still needs paying. I know you need to prioritise but its worth considering if its, say, a trade for a cheaper mobile phone plan. Obviously not if it involves going without something more critical.