I don’t think that’s how people normally do it; partly because I think it makes more sense to try to find good password *schemes*, rather than good individual passwords, and measuring a password’s optimal encoding requires knowing the distribution of passwords already. The optimal encoding story doesn’t help you choose a good password scheme; you need to add on top of it some way of aggregating the code word lengths. In the example from the OP, you could use the average code word length of the scheme, which has you evaluating Shannon entropy again, or you could use the minimum code word length, which brings you back to min-entropy.
I don’t think that’s how people normally do it; partly because I think it makes more sense to try to find good password *schemes*, rather than good individual passwords, and measuring a password’s optimal encoding requires knowing the distribution of passwords already. The optimal encoding story doesn’t help you choose a good password scheme; you need to add on top of it some way of aggregating the code word lengths. In the example from the OP, you could use the average code word length of the scheme, which has you evaluating Shannon entropy again, or you could use the minimum code word length, which brings you back to min-entropy.