You could split the mortgage, put say 50% fixed and 50% variable. Then you can pay down the variable part as you wish and yet still get the security of the fixed part.
Otherwise you can start overpaying the mortgage immediately at €400 per month taking advantage of current low rates but shielding yourself from the shock of higher repayments in the future.
If you are thinking of putting it on deposit, make sure you are disciplined and pay the savings off the mortgage. With two young kids there are unforeseen expenses that may see you dip into savings for non mortgage reasons.
Only fix if interest rate increases will overstretch you, don't try and time the market.
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wohoo 1000 posts - it's the little things that make you happiest!