You’re right, there is a bylaw restricting riding to only a few Chamonix trails in July & August, however there is no bylaw (well, different bylaws, but easier to get round) for the trails at either end of the valley, Les Houches and Le Tour that lead into different communes or indeed countries.
As you have a car you’ll be even better served as 45mins to 1hr driving gets you La Thuile, Grand Massif, Portes du Mont Blanc, Verbier and other smaller places. You could even go to Les Gets if you had to….
Chamonix trails ARE busy during those months with walkers, so for a pure mtb trip there are better times to visit, but with kids who presumably won’t want to just bike all the time, and the flexibility to travel about a wee bit, you could do a lot worse.
If you want more inspiration for trails, check out: http://www.chamonixbikeblog.com/ (though maybe look at the posts in the summer rather than the last few with snow on the ground, click the “bike ban” tag for rides you can do whilst it’s on, I’m sure you’ll find plenty to keep you busy…..)