How to change the document language in Keynote and make the red lines disappear?
Turns out I figured it out not long after typing this question! I just thought I'd experiment a little and found out how to do it.

You change the language in Edit > Spelling > Spelling... and then reopen the document. That did it for me. I'm guessing it is a bug: I don't think having to close and open a document for language error display settings to make effect it's very intuitive, but I'm glad it can be done! 

I'm posting this anyways to share the knowledge and in the hopes it will be useful to somebody else, struggling with those dotted red lines!