The way I got it to work in the end was pretty hacky.
I created an inline widget called Image (Security "This widget can be used as inline widget").
This widget had an Image field in it (Data type: text, form control: Media selection) and outputted the selected image. I added the 'InsertWidget' button to the richtext toolbar. This way I could get images to display in richtext output (although do be aware that widgets render in the rich text editor as a cog icon button).
Note: if you do this make sure you Resolve the inline controls properly whenever you render out the richtext