Is 'Invoice' no longer a valid transaction type to apply custom transaction body fields to?

If you apply it to a "Sale" then you can apply it your invoice form.

So the field I'm trying to create on the Invoice is a 'ship complete' field, which is native to the SO I believe. Can I apply the native field to the invoice, or would I have to create a custom copy of the native field in order to apply it to invoice?

Did you try to edit the invoice form to see if the field is visible on the form?

I don't see it under the various subtabs of the 'Screen Fields' tab when customizing the form, no

You can create a custom field and source it from the SO.