>>> . . . but it has a long number on it anyway so surely that would be proof of having a one?
Not unless they've got access to the database that lists all currently valid cards, which they almost certainly haven't.
As I read it, the only way to get online discounts using a Blue Light Card is to go through the card company's own web portal:
"As part of your application to join the Blue Light Card community and purchase your personalised Blue Light Card (£4.99), you will need to go through some simple eligibility checks to confirm that you are a member of one of our services. Once confirmed and having received your card, you'll be able to use our online portal, free mobile app and your physical Blue Light Card to get discount codes for a range of online offers, high street retailers, restaurants, cinemas and more!"
https://www.bluelightcard.co.uk/contactblc.php