Just my penny's worth, based on court experience of many, many such applications for leave to remove for travel.
If the mother agrees, get it in writing, agree the dates, times of flights, outgoing and return, where the kids will stay, and the cost of the tickets, the date the passports will be handed over so you can book and the final date at which the mother can object, or be subject to refunding you the cost of tickets/accommodation etc. Use a mediator if necessary.
Once you have the signed agreement, and the tickets are booked, if the mother changes her mind, you are set to apply to court for an SIO on an urgent basis and claim any costs you lose due to her changing her mind.
You will be very likely to get the SIO as it is a big family reunion - the kids are old enough to travel for four weeks with their father, if they want to go. At 10 and 12 they should have a good grasp of how long 4 weeks actually is.
Parenting separately is about compromise and what is in the kids' best interests, not one upmanship. You may be overthinking this as you imply you have a great co-parenting relationship, so there really shouldn't be any reason the mother would agree and then change her mind. Talk to her, not a bunch of strangers online.
Don't assume the schools won't agree. If your kids attendance is excellent, then they are more likely to agree, get their agreement in writing as soon as you have the mother's agreement, and before you book tickets. Travelling does, after all, broaden the mind, and most parents having home schooled their children during the pandemic can cope with a couple of weeks of home schooling so the kids don't fall behind.
It will be a one off - so in my experience both the school and the courts will be sympathetic to that.