thanks for the clarifications! So if I understand correctly, you determined alpha and n for different fit intervals and that’s how you got the values for the first and second plot?
thank looks like a very good fit actually! That you could get the same result for mean within statistical uncertainty for different values of alpha and n shows that the fit is very stable.
As far as I understand now, what you call “fit-dependent fluctuations” are essentially systematic uncertainties related to the choice of your fit interval. You have very small statistical uncertainties, so naturally it’s very challenging to get systematic uncertainties that are just as small. I think that they are in the same ballpark just “beyond 1 sigma” is already great.
In your case, I would leave it at that. If you’d tell me that your systematic uncertainty is smaller than 0.01 % I would not believe you anyway 
(…okay maybe if you have a really good physics motivation for the choice of the fitting shape).