Thanks to Diglin for pointing me in the right place. Just to present the answer properly: You can do this by using the addComment method, ...
c) If I switch to the SOAP API to raise the invoice - at times, I may need to invoice different items to those ordered. E.g. If a customer orders an item ...
... to update the order status (AKA add a sales order comment) via the REST API. You have to use the SOAP API and version 2 makes it easiest.
Try this params. It is working. { "entity": { "entity_id": 1, "state":"processing", "status" : "processing" } }. Check mapping in sales_order_status_state ...
you can not change order state with magento api. So you should create invoice. see here.
Edit on GitHub
How to create an order status?
Also, update the order status in sales_order table columns state and status.
If you look at the swagger api documents (either on your local machine at / swagger or at the devdocs). You should find the API V1/orders .
All orders have an order status that is associated with a stage in the order processing workflow.