Hi,
I’ve been looking into this lately. From some emails with Taxamo, and also reading through their API docs, it’s my initial impression that you don’t really need to integrate their API directly with your checkout. You can use their API at the end of each business day to process all your EU orders that your shop received. Basically as long as you inform the Taxamo backend about the EU orders before they ship it looks like it would be fine.
The only reason (that I can see) to prefer integrating Taxamo with the checkout would be to get the calculated VAT on each order. But Shopify already calculates the VAT on EU orders if you provide an IOSS number in your store settings.
The above might be a decent solution - I haven’t tried it yet. The downside is having to remember to put every EU order through their API semi-manually.