My understanding is that you shouldn't pay two lots of VAT?
If I'm a UK dude I should be paying UK VAT on anything I buy, regardless of whether I'm buying it from a UK or EU outlet.
If I'm an EU dude I should be paying EU VAT on anything I buy, regardless of whether I'm buying it from a UK or EU outlet.
So, if you buy direct from BKP, the BKP price to you should drop by 20% and, if you purchase, you owe your government VAT at their prevailing rate.
(As an aside, a little while ago, a year or so, I thought the BKP prices looked very low on the website one day. I found out that was because the website incorrectly thought I was outside the EU/UK (this was before we were out), and so it was quoting me ex-vat prices. I refreshed the website and the prices went back up to usual ones I see, including UK VAT.)
How it all works I don't know exactly, but I have looked at it a little bit from a UK customer perspective buying direct from an EU outlet...
I was looking at an amp on Thomann earlier. They have big warnings up that, as a customer in the UK, they are now displaying prices to me excluding VAT for big ticket items. If I purchase, they will take the excl VAT price from me. On a big ticket item, the UK VAT will now be collected from me by the shipper (UPS I think it said) at the UK rate of 20% and the shipper will pay it to UK HMRC (and deal with the paperwork) as they bring it into the UK. Thomann are also warning that the shipper may charge a fee for the VAT paperwork, they don't know how much but guess £5 or so, and their shipping fee, obviously. I've never bought from Thomann, but I assume the price quoted a few weeks back would have included EU VAT and at the checkout they would have collected the shipping fee upfront and paid the shipper themselves.
For items below a threshold (I think it's £135?), Thomann is registered for UK VAT, and they charge the UK customer themselves and pay it to the UK's HMRC.
I understand these procedures were all part of the deal hammered out to ease moving stuff across borders and enable tariff free trade.
I believe something similar should happen in the other direction (UK > EU)?
I can't find anything about it on the main BKP website (but I'm in the UK) - it might be worth contacting them to find out?