To answer your question, Pepper is the only lender who might give you a loan based on a mortgage free property.
But you should not do this. You are paying tax on the rent that you receive from your Cork property. You will get no tax relief on the interest which you pay on the Dublin property. With an income of €24k, €400k is to heavy an investment in property, even if the mortgage is only half of that.
It's also risky. What happens if the tenant in Cork stops paying? You will be completely ruined. Your credit rating will be destroyed. You might even be forced into a fire sale of one of the properties.
Sell the house in Cork. Use the proceeds to buy a house in Dublin. Live a simple financial life. No mortgage. No rent. No tenants. No RTB. No tax. No hassle.
When you want to move back to Cork, sell the house in Dublin and buy a house in Cork. Alternatively, as you will have no rent, maybe save up for a deposit to buy a house in Cork a couple of years before you are ready to move back.
If you are planning to move back within the next 5 years to Cork, then you should probably keep the house but you should not buy a house in Dublin. Rent until you are ready to move back. If it's more than 5 years, then sell in Cork and buy in Dublin.
Brendan