Translation to ACF relationship field

You should leave the post type name (and everything else that might need to be translated) in English.

Then, in WPML, you can translate its name (labels, and slugs).

If the main language of the site is Danish, you simply set this language as default language (not for WPML String Translation though, where I would keep English as default).

If the reason of using WPML is to just provide a Danish back-end, but the site is not supposed to be multilingual, the way to do so is by enabling a second language (e.g. English) and marking it as hidden.

But still, is always better to keep all strings in English (gettext, custom post types labels and slugs are strings).

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