Mesher orders are increasingly rare, but I do think there is a good case for one here given the children's ages.
I don't think you can reasonably have 100% or even 80% of the house full stop. Particularly as the higher earner, with a higher pension. Even if he agrees (which sounds unlikely) I don't think you'll get a judge to sign it off. You might "need" the house for the next couple of years while the eldest is doing exams, but your housing needs are equivalent to his (in the eyes of the law) and will drop significantly in a few short years.
But the more I read, the more it seems that there aren't enough assets to house you both (with room for the kids) appropriately at the moment.
Currently my understanding is eldest is doing GCSE's youngest not at important exams yet. One option could be mesher until eldest is 18 and then you and the younger move somewhere cheaper.
Understandably you want youngest to stay at this school too. It sounds like Ex wants that too. And it works in his favour because once youngest is 18 your needs drop dramatically.
The only problem I can see with a mesher is that if him and his partner split he'll have problems housing himself and the kids in the interim. He could also argue that he's staying there in the short term but needs to house himself and kids appropriately with the divorce settlement.
A sensible proposal could be. You stay in the house until youngest is 18. At that point, home is sold and you split the equity. I'd expect that to be 50/50. You'll probably have to equalise pensions.
Also worth considering that you can access tax-free lump sums from private pensions at age 55. That means that you both have half the home equity plus a lump sum to play with if you want to live somewhere nicer.
At that point the money will be sufficient since you won't be tied to schools, and could move somewhere cheaper and with fewer bedrooms.
I think you need to let go of this bonkers 80% to 100% idea. It's just unjustifiable, particularly given the ages of the children.
If you want to keep the house, you could consider giving him your pensions to the value of 50% of the equity. That kind of depends on the numbers. Your pension will take a massive hit, but you'd have 15 years to try and make up the shortfall.
If your pensions are DC and he's also around 50 he could access the funds (and 25% tax free lump sum) in just a few years. It's generally not advised, but if the house is the be all and end all it's an option that achieves similar outcomes to the mesher.