A UK VAT-registered business must NOT charge VAT when selling to customers within the EU.

You've applied the rule too generally.

The rules are:

  1. Customer is an EU and VAT-registered business: no VAT *if* proof of business provided in the form of a VAT number.

  2. Customer is an EU business: no VAT number: charge VAT at your local VAT rate.

  3. Customer is an EU consumer: charge VAT at *your* local rate.

  4. Customer is based outside the EU: no VAT.

The new rules affect the above thus:

  1. No change.

  2. Charge VAT at *customer's* local rate.

  3. Charge VAT at *customer's* local rate.

  4. No change.

The new rules also place a significant burden on proving the customer's location: you must verify their address using at least two methods, one of which may be the address the customer provides; and the other may be their credit card address, or geoip lookup via their IP address.

Whichever proofs you rely on must be recorded with the purchase, and kept for 10 years.