Living items can't be sent through the Chron-O-John, but there are ways of preserving living critters so they can be woken up in the future.
Have you ever heard of freezing people (or hamsters)?
There's a way to freeze the hamster in Bernard's time.
Drop the hamster in the ice machine and have Laverne take it.
Now all you need to do is thaw the hamster so he'll wake up.