I hope you don’t mind if I just focus on the language-learning aspects, and leave it to others to discuss the part related to Clozemaster.
(1) French already has a very high concentration of English cognates. I’m not sure that developing an “enriched” list is worth the trouble. It does sound like a lot of effort on your part.
(2) The list of cognates you linked to includes some rows like these:
Cake n. Bolo (Portuguese) Torta (Spanish) Gâteau (French)
While people do talk about tortes in English, I don’t think “bolo” is a cognate at all, and “gâteau” is only a cognate in a convoluted sense. The Online Etymological Dictionary has this entry for the English word “gateau” (which I have never heard used in English):
1845, from French gâteau “cake,” from Old French gastel, from Frankish *wastil “cake,” from Proto-Germanic *was-tilaz, from PIE *wes- (5) “to eat, consume.”
Unless your son speaks Old French, Frankish, Proto-Germanic, or Proto Indo-European , I don’t think “gâteau” is going to be any easier for him as a beginning French learner than any other word. So you probably would have to filter that list to remove such words.
(3) When one learns from a skewed collection, there’s always the possibility that one’s knowledge and skills end up getting skewed. Some of that is not a bad thing for a beginning learner. But saving the noncognates for later could just mean that he hits a wall when he encounters them. If there are noncognates that are important for building everyday sentences, this will only delay the point when he can do that.