You can and should still use hv005 as the sample weight, but doing your analysis with smaller geographic units is potentially problematic. The sample is designed to be representative at the region level, but not at the level of smaller units. As you disaggregate the data to smaller units the sample is less and less likely to be representative. The sample is also designed to provide a certain level of accuracy at the region level, and again as you disaggregate to smaller units the accuracy of those estimates gets worse and worse and the confidence intervals around the estimates quickly become very large and unreliable.

[inference about the] district location of the sampled clusters using a GIS software and the GPS dataset these data would not be statistically representative.

cluster-level measurements are based on too few observations to be meaningful in and of themselves - as you say, there are wildly under-powered. A couple of things you could do: a) by averaging over many clusters, you can still get good estimates of community level variables, but each individual cluster-level point-estimate would be very, very noisy. But they may still mostly "agree" in some sense; b) so if in your hierarchical model you allow each cluster an unconstrained cluster-specific effect (like treating each cluster as a mini-experiment), you could look at those individual point-estimates on a scatter plot (say Beta across some variable you think would affect Beta); c) and then you could start restricting those Betas to have some particular distribution (a random slope model) and see how that changes your overall point estimate as you make your priors on the distribution of Beta more/less informative. I think this makes sense as a kind of model-checking or informal/additional inference procedure. A leave-one-out cross-validation approach might make sense too, depending on how you end up thinking about each of these within-cluster estimates.

Norms are defined at the cluster level, which serves as our community-level unit of analysis

For example, researchers may consider including in their models the average education within the sample (cluster) of approximately 25 women interviewed in each primary sampling unit (PSU). However, this is only a proxy for the theoretically more interesting average among all women in the PSU, and, in principle, the estimated effect of the sample mean may differ markedly from the effect of the latter variable. Fortunately, simulation experiments show that the bias actually is fairly small - less than 14% - when education effects on first birth timing are estimated from DHS surveys in sub-Saharan Africa. If other data are used, or if the focus is turned to other independent variables than education, the bias may, of course, be very different. In some situations, it may be even smaller; in others, it may be unacceptably large. That depends on the size of the clusters, and on how the independent variables are distributed within and across communities. Some general advice is provided.