It is feasible, and I have done it. I was there on a September day and, being somewhat of a weather geek, I could tell that the overcast and light rain in Paris would likely depart by mid-day. I also theorized that most tourists who were not weather-aware, would stay away and try to do indoor things. It paid off, I just showed up on a cloudy morning and bought walk up tickets and the elevator for the top level with a very short line (other than security) and as I climbed to the second level, the clouds broke and I was rewarded with stunning sunlit views of Paris.
That being said, if this is on your kids "top of their want to do lists," I am not sure I would count on that. I had been up many years previously, and was prepared to walk away if it didn't look good. And I was there in September. July, and the particular "given" day could affect your chances. If it is that important, this may be a reason to pay the premium for a reseller. I have also read that there are "behind the scenes" tours that includes access to the top—but I have no first-hand experience.
Good luck.