Property 1: About €350k outstanding, interest and capital. Mortgage about €1200, rent €1050. Probably worth €200-250k at a push.
Property 2: About €220k outstanding, interest and capital. Mortgage about €1100, rent €1000. Probably worth €125-150k at a push.
Without knowing the term and the interest rate, this is not easy to analyse.
We also need to know who the lenders are as the policies of the lenders differ.
I am guessing
Property 1 is a tracker on ECB + 0.5% for 30 years - or else it's a BoS interest only mortgage.
Property 2 is a SVR at around 4.5% for 30 years
Assuming Property 1 is a cheap tracker, it's enormously profitable for you. You are getting €1,000 rent and the interest is €300 per month. The difference of €700 is capital which means you are knocking €8,000 a year off the negative equity.
Property 2 is at least breaking even.
I think you are better off holding onto both properties and hoping for either or both of the following:
1) Property prices rise and eradicate your negative equity
2) The lenders offer a deal to exit the cheap tracker.
You have €570k in borrowings and €370k in property.
Even if the bank agreed to give you a third mortgage, you should decline it. Even on your very high salaries, you are overexposed to borrowings and property. You are vulnerable to interest rate rises and property price falls. You should not increase this exposure.
You should build up a pot of savings instead of paying capital off your mortgages
This could be useful if a lender offers for early repayment of a tracker. I think it's unlikely that such deals will be offered, but be ready just in case. (This is yet another reason for not committing yourself to buying another house at the moment)
If you have savings, you might be able to trade-up - see below.
Your lender might offer you a trade-up mortgage
If property 2 is a SVR mortgage, the lender might allow you to transfer the negative equity to a new property. In other words, sell this property and buy a new one and move the negative equity with it. This would increase your overall exposure, but not by as much as a third property and mortgage would.
If either of your mortgages is with Bank of Ireland or Ulster Bank...
They both allow borrowers who meet their lending criteria to sell the existing property and retain a tracker mortgage at a slightly increased rate for 5 years. Worth considering.
But provide all the information, and you will get more comprehensive options.