I don't own a Mac so any opinion I have is suspect, but it does seem clear that something in the Mini is "sleeping".
What I see from that log:
- Calibre is awake enough to respond to CC's connection request.
- Calibre starts the sequence of "jobs" to actually connect.
- Calibre sends several "commands" to CC, and CC successfully responds.
- Calibre successfully sends CC a command with information about the calibre library.
- Calibre does not receive CC's reply to the library info command until the Mini is "tickled".
From CC's point of view, calibre went away in the middle of processing the library information command, then came back. I don't see any obvious reason why calibre would pause after sending that command. The networking for that step is exactly the same as what happened successfully in previous steps.
One possibility is that the calibre GUI is being blocked. If this happened then the wireless device networking thread would also be blocked. Various GUI things do happen during connection and CC does continue once the GUI is awakened, so there might be something here.
Getting a calibre debug log might help because it will show what calibre thinks is happening.
This FAQ answer explains how to get that log.