I finally managed it after a long time of searching. And it’s pretty simple.
I can grab the contents of the WYSIWYG field like this:
tinyMCE.activeEditor.getContent( content );
And set them like this:
tinyMCE.activeEditor.setContent( content );
I haven’t managed to get the wysiwyg field to trigger and event but I can work around that.
Also I don’t know how this would work if there were more than 1 wysiwyg field in the form.