You’re just into the bike ban time at the start of July, but fortunately the trails are still relatively quiet then, nothing like August when all but the most hidden aren’t worth riding as you’re constantly stopping (and chatting, on the whole walkers are less bothered by bikes over here)
If you’re just in town for a couple days then don’t worry about the ban anyway as there are still a load of legal trails that will easily keep you occupied. Having spent 6 months in L2A and 6 years in Chamonix I’d also say that there’s far more to keep you occupied away from the bikes in Chamonix! Yes, there’ll be big queues for the Midi, but the rest of the valley will be fairly chilled out.
For more inspiration (and something nice for the coffee table) check out Tom’s bike book ( http://www.thechamonixbikebook.com/ ) which includes lots of trails for the bike ban and the blog ( http://www.chamonixbikeblog.com/ )particularly the pages on bike ban legal trails here