Even if they would lend to you, and I'm not saying they would, do you really think it'd be a good idea?
Assuming you have no investments other than property (I get that impression from the post), you'd end up being very over-exposed to property. Interest rates are rising at the moment and you could end up in a very messy predicament in a year or two.
Your 1,100 savings per month, is unlikely to cover the mortgage and expenses of even one of your rental properties if it happened to go empty for a period of time.
Taking the above into consideration, if you were to secure a mortgage and buy a third property, you'd be struggling if one of two rentals went empty for a few months - as well as the obvious struggle if job loss were to occur or some other life event - such as another child.
Add to this the fact that any rises the government introduce with regards to property tax will hit you three-fold and the fact that all it takes to cause widespread drops in rents is for the government to reduce benefits to the extent that immigrants are likely to leave Ireland and return home.
There are far too many variables that could leave you in a very tough position with your plan of a third property. Now, I'm single, which means that I don't have children to worry about - but three properties would be above even my level of risk tolerance (I'd consider a second property if my total equity across both exceeded 25%).
With this in mind, what would I do if I were you? I'd pay all but three months worth of living expenses from the 160k towards the mortgage in your own apartment, bringing it out of negative equity and reducing your payments significantly.
I'd then keep the three months living expenses in an instant-access deposit account and use the savings in monthly repayments (that come from your reduced mortgage) and the current 1,100 monthly savings to rapidy overpay the mortgage.
I'd put your current place on the market and start looking at your second-stepper properties now. By the time you find an ideal property, your mortgage should be at a level that, on selling your current place, you would have enough left over for a 10% deposit on the new place.
I'm not sure of your reasoning behind wanting to keep your current place - perhaps it's the significant falls in prices over the past few years. If that's the reason, just remember that the place you're moving to is likely to have fallen the same amount, if not more.