Depending on the wood species, some bark is quite resilient to come out, other just peal like bananas or brake apart when worked on. I'm not sure in you want to make the clock using a slice of the log(wood) or using some other shape but, very thin CA is your best friend in this case but, use the CA to soak the best you can the joint between the wood outer layer and the bark inner layer. If separation is to occur will be the bark separating from the wood.
Another thing that will help to keep the bark together is to spray/brush it with varnish, this is normally done as a final finish, but can be done before end if bark is to "flaky".
Good luck!
Cheers
George