Why can't I save a Send HTML Email action?
Thanks Pierre for your suggestion. I didn't need to try it, but it makes a lot of sense. Wish I'd have thought of it!

In the end (before I saw your suggestion), I rebuilt the email using a different generator tool, which seemed to produce much smaller and cleaner HTML. 
This works OK. It looks like there was either a problem with the HTML or there is some kind of limit on length.