If the shiitake log has been completely spent by the mycellium it is by definition already ready for most species of stag beetle once ground up. This is essentially white rot wood. This is how the breeding logs they sell in Japan are sold. These logs are capable for rearing most stag beetless. This is also how the mushroom substrate they sell is created, it is a byproduct of shiitake log production. In Japan they sometimes it without being composted further, and it is suitable substrate for several stag beetle species. For rhino beetles, you will definitely have to compost it further. It would not hurt for you tor ferment it further for your stag beetles either. A word of note though, not all stag beetles can survive on a white rot wood only substrate... but that should only really matter if you happen to keep exotics somehow. Think of it like kinshi that has been accidently fruited. It isn't as good as kinshi for stag beetles, but the lignin has been nearly completely broken down though.