Have you checked with National Irish Bank? I can't see on their website any mention of a minimum, although they may have one.
In any event use Karl Jeacle's Mortgage Calculator to figure out whether or not the cost of re-mortgaging would be justified over the remaining term of your mortgage if you switch from 5.4% to National Irish Bank's 4.5% LTV Tracker rate.
Could you switch (remortgage) to NIB LTV at 4.5 increasing the mortgage to 75000 or whatever the threshold is.
Then once the account is set up pay the 40000 lump sum on to the mortgage account.
Set up the initial term of the remortgage to give you the payments you want going forward.
I use NIB online banking to transfer lump sums to my mortgage account and it works very well.
How long is left on your mortgage ?
Going to NIB at 4.5 % would be approx 18 euro a month less (if it is a 25 yr loan).
If there is only a few years left on the mortgage- just try and pay off some of the capital and save on interest that way (if you lender allows you)
Click into loan to value mortgages lead the way, read more, then it’s flexible, go down to terms and conditions it says €20K is the minimum. It also says terms and conditions apply and valuation fees may apply, so cant see it being worth your or their while for €35K.
I reckon your best way to save some money on this is to up your mortgage payments (if thats possible) and pay off the mortgage early....assuming your mortgage will aloow it without penaly.
As it stands you will pay about €9500 in interest over the next 10 years. If you pay it off in 8 years, you'll save €2000.