Following is a response from DHS staff member, Tom Pullum:

The estimates are unbiased at the district level but DHS would say they are "not representative" because the standard errors would be so high.

What you want to do, looking at variation across districts, sounds good, and you can do that even if you do not have names for the districts. But you have to be cautious about over-interpreting the extreme values. Statistical instability because of small numbers of cases in many of the districts can lead to exaggeration of the estimated range across districts. You could google something like "Bayesian adjustment for hot spots", for example, to find ways adjust for this effect.