What about using some of the savings to pay down short term debt, and apply for a slightly larger mortgage - you haven't said how much of the 50k was intended as your deposit.
What is the cost of the House/.
What % rate are your car loans at and how much per month are these repayments?
Can you show proof of rental payment over the 6 year period.
Can you show how much you have been saving consistently over the past 6 years.
I am a bit puzzled that you much borrowed so much for the cars while having savings.
Consider that if you paid off the car loans that you could put more per month into savings.
Have you considered the other institutions or a mortgage broker.
I would try and show the next lender that you have the ability to pay by reducing or eliminating the car loans even though you might have less savings,
have proof of your rental payments and proof how you have saved the €50000. I know that your girlfriends contract situation is a bit of a minus but that is very common nowadays.
I know from personal and friends experience that Aib are very negative on lending to any sector. A lot of the excuses for refusal are tailored to each case but do not make much sense.
Maybe I'm missing something, but why do you have 50k in savings and 20k of debt. You're most likely paying twice as much interest on your debt. I'd pay off the debt as soon as possible, whether you're buying a house or not.