Hey Adam!
I'm glad to hear that you're going to be going on the Unlimited plan-- it has a lot of really awesome features that I'm sure you'll make great use of.
However, with regard to the use of the Royal Mail app and Better Shipping, I'm not too sure how these two apps would talk to each other. It might be best to reach out to Atom Yard to see if they have any insight here, as they support their app. You can reach out to them at better.shipping@atomyard.com
Matthew Buttler
Guru
Shopify Inc.