When ever I skip a puppet show it is not put into my inventory and I have to contact support to have it put back. I have tested this bug on all my pirates and it works on all of them. I think that you should be able to have a puppet show even if you skip it because it cant be unfair sometimes
Sometimes the puppet shows don't show up straight away. You often have to finish the quest you are on before it is available in the puppet shows menu. Next time it happens, try playing a bit longer and see if it appears.