Creating external mini-stores for clients

[Jaypex]

Hello! I run an clothing manufacturer working with many small brands hosted on our main website, however many potential larger clients are wanting something a little more bespoke and unbranded. What we would like to do is build a mini-store template that we could change branding / domain and list products from our main Shopify catalog with orders using our shipping prices.

Here is an example:
https://wearenations.com/collections/invictus  (On the main storefront)
&
https://invictus.wearenations.com/ (The 'mini-store')

Is this something that is possible with the API and where should I be looking? 

Thanks for your help!