The bootstrap on clusters, using the coarsest level of clusters, is valid even if there is clustering at finer levels within each cluster. That's because the bootstrap doesn't make any assumptions about the internal structure of each cluster. It assumes only that the clusters are exchangeable, so that a simple random sample of them (with replacement) is a way to generate data from the right distribution.
Assuming exchangeability, whatever the complicated internal structure of the clusters looks like, the complicated internal structure of the bootstrap resampled clusters will look just the same.
There are multilevel bootstraps used in some contexts for survey data, especially if the exchangeability assumption is problematic because of stratified sampling and finite population size. One example is given by Preston (2009) and implemented in the
survey package for R. I don't think it's needed in your context, though.