Personally, I wish all goods transactions in Europe were with VAT and the VAT was claimed back through the local vat office. The systems a complete mess right now, and HM Gov prosecuting the 80% of businesses that go out of business as though they are all attempted carousel fraud is just nonsense.
(Many of those traders are missing because they failed to get the money from the sales fast enough to cover their purchases and went out of business.)
Each invoice would contain the vat number and vat jurisdiction. The buying company would claim back the vat from their local office, by filling in the box for the sellers jurisdiction.
e.g. I claim back 100E from France for goods imported from France, the UK office would get the 100E from France.
The local vat offices would coordinate the transfer for refunds. Random invoices would be matched up between buyer and seller vat offices to check for fraud. So periodically the VAT office would demand the paperwork for the France invoices and pick a few invoices to check, and the French would match it up with the French sellers to confirm the invoices.
Another thing I'd like to see, is the removal of vat registration for small traders. A company should be able to set up and trade without the burden of VAT for the first 100k Euro, so they can get their cash flow sorted out and accounts before they have to jump the VAT hurdle. I know this is how it is in the UK, small companies don't need a vat number, but in most of Europe you need compulsory VAT registration to sell goods or services, and they refuse the VAT number of all sorts of spurious grounds to block Polish plumbers etc.
That would help small companies get off the ground.