Right now I'm generating 100 Beta RV's and using these data to test the null hypothesis of whether they are uniform, but my Kolmogorov-Smirnov p-value fluctuates greatly (from far below 0.05 to far above 0.1) every time I run my code. Is this because 100 RV's are too small for a consistent answer?
It is usual for P-values to fluctuate because P-values are dependent on the data. Every dataset will provide different levels of evidence against the null hypothesis and so the P-values will differ.
If the null hypothesis is true then the P-value should vary as a random variate with a uniform distribution between 0 and 1. If the null is false then the P-value will still vary, but it will tend to be smaller with increasing disparity between the 'true' state and the null hypothesis, and with larger sample. Yes, a smaller sample will yield more variability of P-value for any given disparity.
It is convenient to think of P-values as a data-dependent random variable.