Fully agree, to me the tests seem simply stupid!! I certainly wouldn't want to pay if they simply say no afterwards. Any chance of asking them what results they are looking for, then an informal assessment from someone who carries out the tests to see whether your children's results will be good enough before doing the formal tests? Or could you simply check that they are looking for results of normality within a spectrum, rather than a pass mark? Maybe it's the school's way of making sure they do not have children with special needs. Our dyspraxic/autistic son's results showed him exceptionally bright at some things and crap at anything to do with motor skills, maybe it's the difficult child they are wishing NOT to have, so if your children are without special needs, then perhaps that's all the school is looking for.
No idea where the better schools are in Lille, communal ones have intake according to strict catchment area, the the posher the area, the posher the school I suppose. You can opt into the very cheap Catholic system instead. I'm in Belgium and the Catholic schools are free here, unlike in France, but they make up over 50% of schools and are seen on the whole, rightly or wrongly, as more prestige schools than the communal ones, perhaps someone in France could fill you in on whether that's a generalism for France too.
All children legally "resident" in France, plus asylum seekers, have the right to an education there, it's not dependant on which country taxes are paid in or nationality. There are thousands of cross-border workers in the EU now, paying tax in one country and living in another.
EABJM Lille is a part-subsidised French state school, the rule about attendance should be the same as in a subsidised Catholic or Communal school.
If any of this is wrong, hope someone comes along soon to correct me.
Hope your children have a great time. I really like Lille, I have worked there on and off since 2007. Great shopping and nice Flemish architecture in the old town, great transport system, great to go anywhere else in France too.