There are two reasons that a lender looks for savings as part of a mortgage application.
(1) The deposit. Maximum mortgage for a switcher is 80% of the property value. If the value of your home was, say, €245,000 then the maximum 80% mortgage would be €196,000. If your Bank of Ireland mortgage is €200,000 then you'd need to put up the other €4,000 from your savings, plus switching costs.
(2) Affordability. Lenders will stress-test the proposed new mortgage to see how you would cope if interest rates went up a couple of percent. Use one of the free mortgage calculators online to see what the monthly repayment will be on your new mortgage. Then add 2 or 3% onto the interest rate and see what the repayment would be. That's doing your own stress-test. Assuming that the stress-tested repayment is higher than your current Bank of Ireland repayment, then you should be saving a minimum of the difference into a separate savings account. That way, you can show the new lender that you could afford the mortgage repayments even if interest rates rise by 2 or 3%.
It doesn't matter where you save. Just don't dip into the savings account between now and when you're applying for the new mortgage. Ideally keep a separate savings account, that shows money going into it only and not coming back out.
Reading your original post, I'd guess that (2) was the reason that KBC rejected your application.