No - there’s no way to support all the anti fraud mechanisms of the major credit card networks without a thorough paper trail. Masked card services help prevent unwanted charges and inconvenience for the customer - they may give a fleeting layer of privacy between the consumer and the merchant but nothing more than that.
I wish more services supported this, but I understand it adds a lot of hassle for them as well.
I don't know what the rules are for sellers that are inside the EU, but if they are at all like the rules for sellers outside the EU selling to buyers in the EU they are required to collect two pieces of evidence that support their determination of which country's VAT to collect.
Where I work we use the country the person claims they are in from the country drop down on our cart and what country MaxMind says their IP address is from. This works most of the time. If those don't match we look up the first 6 digits of their credit card to see what bank issued it and see what country that bank is in, and if that matches either their selected country or the IP country we go with that. If the bank is in a third country, we look at their email address and if that is at a service that is mostly just serving one of the three countries we go with that.
How would a company that accepts cash and keeps very minimal customer information deal with this?