It’s worth noting that the BayesFactor software itself does not use the Savage-Dickey identity anywhere to compute Bayes factors. It uses a simple Monte Carlo integration, importance sampling, and/or quadrature (with various methods of choosing between them). Most of the parameters, in fact, are analytically integrated out first, which is possible due to the conjugate framework we use.
Regarding independence and improper priors, I agree that independence is not the right way to think about it (since these aren’t proper probability distributions). The key is the desired transformation invariance of the inference, not independence.

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