I want to order VAT-free by providing a billing address outside the EU and a shipping address within the EU

This isn't possible. This is due to the rules for Value Added Tax (VAT) and customs:

• VAT reverse charge: For transactions outside the European Union, VAT is shifted. The law requires that the country on the invoice matches the country where the goods are physically shipped.
• Customs check: Customs checks whether the recipient in the destination country is also the person listed on the invoice. If the countries are different, customs can't calculate the tax correctly.
• System limitation: Our system can't process VAT reverse charge when the invoice country and shipping country are not the same.

What should I do if I still want to use different addresses outside the EU? 
If you live outside the EU and the countries are different, you must enter the shipping country as the invoice country as well. Otherwise, your order can't be shipped.

Does this also apply to the United Kingdom (UK)? 
Yes. Since the United Kingdom is no longer part of the EU, the same rules apply. The billing address and shipping address must be in the same country.