Yeah they'll want the business name that sold you the illegal software. Who cares about the key, they issue new ones all the time over the phone. At most, you'll have to fax Microsoft a copy of your invoice proving your purchase. This isn't a big deal because you are trying to be legit, aren't you?
The only way it will back fire on you is if you run your mouth with nervous chatter. Steer clear of the "My new pc is slow, I wanted a disc" jabber and just give Microsoft the short facts. "I paid for a pc, there was no genuine hologram media so called them, they gave me a run around, tried to re-sell me software so I am calling you because something is up over there". And don't give any dollar amounts on what HP said for media. You don't care what it cost and neither should Microsoft. You already paid for it once and you've got the invoice to prove it.