I believe as EU based seller, you're only obligated to include UK VAT to all EU customers. I believe the UK's VAT rate is 20%. So if I would buy something from you online from my hometown Amsterdam, I have to pay $ 100 + UK VAT = $ 120. To all other countries, you can sell that product for just $ 100. So you could setup your products without VAT included and setup your store to add the 20% UK VAT rate to all EU countries. By the way, I'm assuming you're selling B2C. For selling B2B other rules apply.
There's one issue with the above setup, the product price listed will be without VAT. To list 2 prices (one with VAT and one without), you could consider Multiple Pricing. We developed this app for exactly this reason. Have a look, you can try it out for free for 7 days: https://apps.shopify.com/partners/developer-2cf34194a44207d7
Let me know if you have any remarks or question. I'm happy to help and very interested in the topic.