"However, how does one calculate the SE for the 99%, 99.9%, and 99.99% values? It's not the same as the average's SE." - you seem to be confusing stuff here, so let us break it down into smaller parts. "51.244" is what, 99.9% value or your average? - It is your average, so the standard calculation of SE may be used, and there is nothing special to derive.
"There are many different SEs: the arithmetic mean, median, harmonic mean, geometric mean, etc... And as far as I can tell there is no formula to calculate an SE for arbitrary quantiles, and I really don't feel like or think I could derive it myself." - I think you are diverging from what is needed here. None of these are needed, we already have an estimator, 51.244.
"The simulation simply needs to be run multiple times if you want CIs for the 99-99.99 percentiles. I truly cannot think of any other way." - if you are not very familiar with statistics, specifically with Monte Carlo simulations - what you propose will work. And will probably be more failsafe, as if you derive the value via algebra - you are probably more likely to make a mistake. Depending on your computer though, it may be slow. It does not *need* to be done, but it is "a way" (but there are others) of how it can be done.