Since no one has commented, I will make a few remarks.
I do not have a copy of B&A at home, but I believe the Akaike weights were not related to the kind of model averaging that you are doing. As I recall, in B&A, they used the full data set and ran all subsets or regression models. They then assign weights to the various models based on the AIC criterion for each model,
That is different than the MODELAVERAGE statement, which uses bootstrap resamples of the data and then computes an average, which is the bootstrap estimate for the parameter estimates.
Now, nothing stops you from using a weighted average of the bootstrap estimates, but I don't know whether that makes statistical sense.