(posting an answer to my own question) I'm not sure if this is correct but after thinking about this for ages (and based on the answer by @LmnICE) the following seems to make some intuiative sense to me, please comment/correct me if I have this wrong

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I think the confusion is over what the posterior actually represents. Being explicit our posterior as noted in the OP can be expressed as: 

$$
P(\mu_j, \mu, \tau, \sigma \ \mid X) \propto P(X \mid \mu_j, \sigma)P(\mu_j \mid \mu, \tau)P(\tau)P(\sigma)P(\mu) 
$$

That is to say that a sample of $\mu_j, \mu, \tau, \sigma$ from our posterior distribution is equivalent to a sample from the model which assumes these relationships between the parameters. 

Or put another way, the posterior hasn't changed any of these assumed relationships; in fact the posterior is dependent on them. E.g. a sample of $\mu$ from the posterior is equivalent to a sample of $\mu$ where we are assuming it is the log-centrality parameter of a log-normal distribution for $\mu_j$. 

I think the confusion I had is that we often refer to the posterior as being an updated / changed distribution but this is merely referencing the fact that $P(\theta \mid X) \ne P(\theta)$. However under this model the data doesn't change the relationship between parameters e.g. $P(\mu_j \mid \mu, \tau)$  is still $P(\mu_j \mid \mu, \tau)$, all that is changed is what values we think those parameters are. 

So to answer "question of interest" I would argue that as our model is assuming a log-normal releationship that the population-mean of $\mu_j$ is indeed $exp(\mu + \tau^2/2)$. Likewise then the interpretation of the values of $\mu$ and $\tau$ that come out of the posterior are the values that most likely fit this assumed model given the available data.