Still trying to find the best way.
Really, in the field of statistics, it's impossible for us to tell these things. Even with the original Gen V non Masuda Method 1 in 8192 chances, we could get 50 within 2,000 eggs, or 1 within 20,000
True, you can never be 100% sure. But you can use statistics to get any degree of certainty below that.
Let's use Masuda method chances, since most people will be using that anyway. We let the null hypothesis of our test be that the shiny chance is double that of B/W, i.e. 3/2048, and the alternative hypothesis be that the shiny chance is greater than 3/2048. Now, let's say we want to be 95% sure that the shiny chance is increased with the Shiny Charm (more on this later), i.e. we need a 5% significance level.
If we get a bunch of volunteers, hatching 10,000 eggs is realistic.
The distribution of the number of expected shinies when breeding with the Masuda method follows a binomial distribution X ~ B(10000, 3/2048), since the chance of getting a shiny is the same at each attempt and independent of other attempts. Since the probability is small and the number of trials is large, we can approximate this with a Poisson distribution Y ~ Pois(10000 * 3/2048 = 14.648).
Now, we need to find the minimum number of shinies that need to be hatched that would only have a 5% chance of happening under the null hypothesis. This ends up being 22; the probability of 21 or fewer shinies is 95.7%, so the probability of 22 or more is 4.3%.
Therefore, to prove with a 95% confidence rate that the Shiny Charm increases the shiny rate of Masuda method breeding, we can hatch 10,000 eggs and get at least 22 shinies from them.
I'm pretty sure you could get enough people to help, seeing as you run the biggest English Pokémon site; if you can get 20 shiny hunters with the Shiny Charm (not unreasonable), that's 500 eggs each, which is not all that much. You just need to be careful with collecting the data; for example, you need to make sure everyone signs up first and only starts collecting data from that point onwards (if you just asked people to say how many eggs they took to hatch a shiny, you'd be biasing the data because people who had hatched hundreds of eggs but not gotten a shiny wouldn't report in).
Basically, it's a big undertaking, but it's definitely possible if you avoid the pitfalls.