Hi Mrs D
You should not sell your house as you will be left with a shortfall and Danske will pursue it as you can well afford to pay it.
There is nothing wrong with asking them to do a deal on the shortfall in advance, but I think it's unlikely that they will do so. The only deal which you might get would be that they would offer to write off some of it in exchange for cash up front. For example, if your family could give you half the shortfall.
However, your situation is not as bleak as you think it is.
Our joint incomes are 75k, we have a 10k car loan and savings of about 5k. Our monthly mortgage commitment is 1088 against a rental income of 750,
You mortgage commitment may well be 1,088 but the interest element is around 660. That means that you are paying around €5,000 capital off the balance each year.
In four years time, your mortgage will be down to €155k and, if house prices don't change in the meantime, your negative equity will be down to €20k.
If you sell your house now, you will end up with a shortfall of €40,000, and you will probably not be able to buy a house ever again. Nothing disastrous about that, but most people do want to own their own home.
It's possible that Bank of Ireland, or someone else, might buy the loan book of Danske Bank. If they do, you would then be able to buy another home and move the negative equity if there is any at that stage. If you sell your home, you will not be able to take out a mortgage as instead of the 20% deposit required, you would have a €40k shortfall.