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Hello, I am Keith, one of the experts on Just Answer, and pleased to be able to help you with your question.
You should tell your EU customer that you are not registered for VAT in the UK and therefore do not have a VAT Number. Remember that some EU countries do not have any VAT Threshold eg Germany where if you do one phennig's worth of business you must register for their equivalent of VAT.
HMRC may allow a registration, but probably not as you are non resident. If they do it will not open the door to UK taxation on the company's world wide income.
I do hope that you have found my reply of assistance.