Hello,
I propose to build this website on Magento 1.9.*, is an e-commerce platform, that offers all a shop needs: I can show you a demo of this platform, if you want, or you can find information of it on google, also you have a backend part, where you can administrate your products, categories, customers, orders, invoices, and more other.
Also I found a free theme for this platform, look like your attached images, maybe we have to change a little bit, some colors etc.
The delivery time, remains if you don't have extra features for your shop, that magento don't have, but we can discuss and we can see.
Thank you, and I hope you want to go forward with this platform, because is very secure and easy for you and customers to use.
Paul.