You would be better off posting this on the UK forum. There are not many who monitor the US forum who are familiar with Europe.
As soon as you have a single unit in any EU/UK country, you have a VAT and income tax obligation. The threshold only applies for orders sent direct to the customer and even then, you need to include VAT in the price and prepay.
EFN only obligates you for the country you keep your inventory in. Pan-European obligates you for every country (UK, DE, ES, FR, IT, and Poland - probably others)
> But what if I want to use Pan-European FBA and let Amazon distribute my inventory to different warehouses in several EU countries? How does the VAT registration work in that case?
You need to register for every country with warehouses. Keep in mind that the registration process is in the local language. You need to know several European languages and prepare your registration and tax filings in those languages.
> Many thanks for your advice, I am trying to decide whether to go for EFN or Pan-European FBA.
EFN is easier to set up and manage, but the FBA fees are a little higher.