You do not need to adjust the weights, and you do not need to make any changes to svyset. The clusters are nested in the strata. The strata are either all urban or all rural, so your urban and rural subsamples have completely different strata and clusters.

Even if you constructed subsamples in a more general way, for example based on age or education, etc., you would not need to modify the weights or svyset. In effect, the adjustments for the survey design (with weights and svyset) will restore the sample to an equivalent simple random sample (srs), and any sub-sample of a simple random sample is another simple random sample, from whatever sub-population you may identify.