Say I have a (sample) bin with balls colored red, green, and blue. I'm asked if the distribution of the colors is uniform with a required significance level of $0.01$, which means is there a $\dfrac{1}{3}$ chance to get each ball color if I take a ball randomly from the bin.
I want to use bootstrap sampling technique with confidence interval $99\%$. I stated the null hypotheses: the distribution of the colors is uniform. And now I want to choose my test statistic. I'm finding it a little hard since there's three things to take care of: probability of red, blue, or green balls. What I've thought of until now is choosing my test statistic as the total variation distance of the probabilities to get each ball in my bootstrapped samples, from the uniformly distributed sample. I would love to hear any feedback about my thoughts and any ideas to make it better, thanks in advance!
I am not sure what you mean by "bootstrapped" but you seem to saying you intend to look at $$\max\left(\left|\frac{n_{\text{red}}}{n}-\frac13\right|, \left|\frac{n_{\text{green}}}{n}-\frac13\right|, \left|\frac{n_{\text{blue}}}{n}-\frac13\right|\right)$$ as the test statistic from your sample, where the sample size is $n=n_{\text{red}}+n_{\text{green}}+n_{\text{blue}}$. That is not unreasonable, but you may have issues deciding what is the critical region. This distribution of this statistic would tend to be closer to $0$ for a larger sample size if the three probabilities are actually all equal to $\frac13$.
If instead you used $$\left(\frac{n_{\text{red}}}{n}-\frac13\right)^2+ \left(\frac{n_{\text{green}}}{n}-\frac13\right)^2+ \left(\frac{n_{\text{blue}}}{n}-\frac13\right)^2$$ then I think for reasonable $n$ you could multiply by $3n$ and then use a chi-squared distribution with two degrees of freedom (which turns out to be a scaled exponential distribution), perhaps using a continuity correction.
If $n$ is small, then you might prefer a multinomial test.