To get a second home

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My partner has gotten a windfall and I think we can afford it.
We would really like to move to the area where this flat is located, but my ex won't allow it due to our shared child.
We would spend 1/3 of our time there, in preparation for a permanent move down the line.

I realise there are many cons:
Second homes are now subject to extra taxes etc.
They take up housing stock.
Extra hassle and headache of maintaining two properties.

But I am tired of my busy and stressful area and feel I need an escape.

Your child’s father can’t stop you, unless you want to move abroad, in which case it’ll be likely that the family / divorce courts would be involved.
Otherwise, how you spend your money is up to you.

He is a nasty abuser and he will definitely make my life horrible if I try to move. He has PR, so if nothing else, I can't just unilaterally change my child's school

Yep, it's your money - spend it on what you want to.

If you’re going to spend a third of the year there would that be enough to cover DC’s time in school? I would have thought not so DC would need to be at the current home quite a bit to go to school rather than than changing schools will.
However, I am wondering whether losing the stress you feel in your current area will only be cancelled out by the aggro from the Ex!

I’d speak to a financial advisor.

I’d also think about other practicalities - is it in a nice/popular area where you could sell in the future if needed? Is your current home easily sellable once DC is an adult and you could move full time, would it still be easy for DC to visit you? Would this 2nd home mean you holiday in this area only, would you still want a foreign/sun holiday/city break, would DP be ok with only holidaying at 2nd house? Can you afford to fully furnish the 2nd house, pay for any maintenance, employ a gardener or cleaning service in case you have a long period where you can’t visit? Will the distance have any significant impact on the wear/tear of your car, can you take the cost of this?

I’d buy it and airbnb it out when I’m not there.