You got my hopes up but it seems I'm stuck paying UK VAT even though I don't think it's necessary any more.
I've run into this with a few different German businesses, I don't know if I'm right or they are, but it seems like they don't _have_ to charge me VAT. Which is underlined by the huge "import VAT" bill I'm dumped with when I have physical goods delivered to me in the UK.
Are you using the computer for business? If so, you should be claiming VAT credits that neutralize VAT (since it's intended as a consumption tax). If they don't charge you VAT on behalf of the UK, the UK should be charging you the equivalent consumption tax by some other means to keep the consumption tax base neutral.
(But I believe the KYC rules aren't only because of this.)