I got this solved in the end by changing the settings in LeftAndMain.php to this:
'valid_elements' => "*
'extended_valid_elements' => "*
Maybe this is overkill (I suspect that it's only the extended_valid_elements which really needs to be opened up like this), but at least I can now paste the HTML in without all the tags being stripped out.
I still can't actually see the content of the tags in the editor but at least they're now present when I view with the HTML button.