Like you, I am also interested to do Bayesian analysis of multi stage survey
data without using model to incorporate effect of different stages. I was
expecting good suggestion in this regard. Since I could not get any response
from experts, I am trying to extend discussion on this interesting area,
although I have little knowledge in this area.
You can proceed in following way
(1) Draw a random sample by using weight given to you.
(2) Use any of supper population model
a. the normal-normal with known variances, with an inverse gamma for
b. (ii) the Dirichlet (prior)-multinomial (likelihood) to represent extreme
vagueness about the super-population.
(3) You can use bootstrap method for analyzing survey data through
Bayesian approach (see http://3w.eco.uniroma1.it/OB07/papers/aitkin.pdf )
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for future discussion.