OP I've been in your exact position - acrimonious divorce and asshole ex who would try anything to piss me off, no matter what the consequence for the dc.
Here's what I did. Firstly advisethem of the holiday, in writing, by email/text so that you have a paper trail. Ask for consent for a legitimate family holiday, to be given by x date. .
When the consent isn't forthcoming, acknowledge the date having passed and advise you will be seeking permission through the court.
Then fill out the form and submit it. Pay the fee and wait for it to come up.
It was a really straightforward process, and the judge sighed and gave me the permission - it was actually a residency order, not at all what I'd applied for but kids were with me almost 100% and I had the evidence the ex was being obstructive. Do you have written evidence from the police and ss confirming they recommend dc lives with you? I think I'd be applying for the order to be fully updated, but the holiday is priority.
Honestly I saved a fortune applying myself - I had a hugely empowering solicitor who pushed me to DIY as she was so sure it was a black and white case. It took 20 minutes max in court,and my ex didn't even show up.
It sounds like your dc has got the measure of their other parent.