Hi,
I had this discussion with Christian Cox and I was content, but now that I'm at the real situation, I face the problem again (which I couldn't quite figure out how I had the impression in the first discussion): https://toolset.com/forums/topic/default-language-of-cp/
Default language of the site is Finnish and I've added English as second language with WPML. Now the custom post type name is in Finnish (Teokset). For me it seems now impossible to translate with WPML String translation because WPML seems to think the default language of CPT is English. So, what did I miss?
What I recommend that you do is to open a ticket with the WPML Support forum for a more direct approach on how to this or for clarity for what is happening here.