Integrating Square products/inventory into on-site payments

Hi.

I'm working with an organization that mostly sells books at physical locations using Square's point-of-sale system. They are able to easily see what products and inventory they have, and they have come quite familiar with using Square for P.O.S.

Now they want me to rebuild their online store, but they are hoping that it can be integrated (or at least linked) with their Square account somehow, so that their inventory remains synced whether they sell something in person or online. You can make online stores using Square, but...it's a separate page (using a mkt.com address), and this organization wants to keep customers on their own site (and not be redirected elsewhere). They want to take credit card orders on site, and for the transaction to be seamless (yet still be connected with Square somehow).

Has anyone else here done something similar? Any suggestions on how to make it work? I've investigated a bit myself, but have come across a few problems:

1). I'd love to use MarketPress and Stripe (because I think it's a great and powerful combo for online stores), but I don't know how I could tie the organization's Square account into it. There's a lot of API documentation for Square, but it's way beyond my limited programming skills to implement this, or even know where to start. Unfortunately they can't afford to hire a second developer, so I'm wondering if anyone else here has used this capability and could offer some advice (or even say if what we want to do is even possible)?

2). I could also use WooCommerce, and a plugin like SquareUp to WooCommerce Import Ninjas. But that plugin has DREADFUL ratings, and so it doesn't leave me filled with confidence. (The store has to work perfectly). Also, Stripe integration with Woo is an expensive add-on, rather than integrated (like MarketPress).

Then there's the fact that Square is U.S.-only, which is fine for the point-of-sales stuff (which has been U.S. only), but they want to be able to sell to Europe and other places. I can't tell if tying some kind of on-site eCommerce solution with Square will mean it will have the same limitations or not.

If anyone has any experience with making an online store that takes on-site orders AND integrates with a Square account (for inventory syncing), and has any suggestions or advice, I'd greatly appreciate it!!