I read your question as in 'which part of the Belgium summer holidays' is the best time for Disneyland in Paris. In that case, July and within the week, a Monday or Tuesday or a Thursday or Friday. See this page by Disneyland Paris. I guess the earlier in July the better as more countries get holidays in July and most do not get the whole of the month.
That is confirmed by this page.
mid-July to late August are very popular.
If you do not want to go in the summer holidays, try to find a public or school holiday in Belgium that is not a public holiday in France.
Edit: On reading the new version of the question I took out the 'list of national holidays' in my diary for France and Belgium. No joy there, all days mentioned for Belgium are also mentioned for France, (besides July 21, which you with good reasons do not want.)
July 11 is 'de dag voor de Vlaamse gemeenschap' (The day for the Flemish community.) This should be a day of for you, school holiday already for the young one and is early(ish) in July, but you ruled out the summer holidays (which I understand.)
The autumn holiday for the schools in Flanders is rather late in the year, but might work for you: 2 till 8 November 2015. I guess that even if there is a school holiday in France at the same time, you will find the park so much more quiet than in the summer. If you do not like that idea, the next likely option is the Easter holidays, where the Flemish schools two weeks: 28 March to 10 April.
A better option, in my fewview, is to talk with the parents and ask for a day the school of your godchild is closed while most others are still running. Maybe for a day the teachers have to get courses, (study dag'studie dag' in the Netherlands) or for whatever other reason.