Hi Graeme,
What I’d recommend in this case would be creating an invoice e-mail template and using that when making a request to the send_email endpoint, just as described in this forum thread: Emailing Invoices using the Contact's email address. This will allow you to have a minimal request body when sending invoice e-mails, i.e.:
{
"invoice": {
"email": {
"use_template": true
}
}
}
It looks like the use_template option isn’t mentioned in our main documentation, so I’ll make sure it gets added – thank you for helping us improve our docs with your query, and I hope this solution works for you! 
Best wishes,
Ewa