NATO staff get I think category 2 entry into European schools. I don't know much about European schools other than they are hard to get into even as category 1 (EU commission) in the English and French sections - you are theoretically only allowed to apply for a section where you speak the language at home. ES might be an option if your children are young especially, but they only start at 4 years old there, if older, some studies must be in a second language, so may not be so suitable for a 12 year old for example, the level of the 2nd language is very high.
I know there are students funded by NATO at the British School of Brussels, their funding is somewhere around 75% mark and you pay the rest, so at least some the international schools in the area must be funded by NATO. It would depend I think on your posting conditions, you may get full funding for an international school in Brussels. There is a big choice on international schools in the greater Brussels area for under 12s, not so many for over 12s.
Depending on age of your children and where you might live next or whether posting to NATO would have a cance of being extended, local schooling may be an idea. It's a very personal choice. Local schools around NATO are full of children of all nationalities, our youngers children's school for example has around 30 nationalities, in some ways more international that the international schools!
I have no idea what it's really like to live around SHAPE, imagine quite quiet compared to Brussels which itself is not particularly lively. As I intensely dislike driving, I'd never consider settling the children into school around SHAPE then getting on the autoroute to NATO for work. We have 2 families in our street who could have gone to school for free at SHAPE and in fact one of them could have gone to the Amercian military school in Brussels (Sterrebeek) but they chose to live here in Brussels, their dads cycle to work to NATO and the children walk 5 minutes to school. They visit SHAPE for OUR supply of Ben and Jerrys ice cream once every few months and I guess stock up on all the other American products that cost 2 or 3 times the price if bought in Brussels, it's a nice perk but not enough obviously to make them want to live around SHAPE and work at NATO HQ.