I don't have personal experience with the Mont Blanc tour, but have researched it some in recent weeks. You may have already checked it out yourself, but here's a website I found with a good collection of info (when to go, how to get there, where to stay, what to expect, etc), plus links to other sites with similar info --
Re doing the route the first half of July, the above website includes that period in recommending "when to do the route" (see quote below), subject to mountain weather ("unpredictable"), winter snow levels, & "country specific" holidays --
"Many refuges only open for the summer season, so given the unpredictability of Alpine weather, walking during the summer is probably the best option. This means 1st week in July to mid September, although this window may shift depending on the winter snows. It is also worth being aware of country specific holiday conventions. For instance, in France 1st July and 1st August are holiday start dates and 14th July and 15th August are public holidays. Expect rail, road and air travel to be particularly busy at these times. The TMB Ultra mountain marathon takes place at the end of August, so if you want to walk then, be prepared to reserve accommodation well in advance and expect the trail to be busy. We travelled to Les Houches on 31st July, walked 1st August to 12th August and usually reserved our next days accommodation the evening before."