I will keep contacting the test center to see if there are any cancellations, I'd be happy to take it again if I can. Calling again and getting some kind of reference sounds like a great idea too, I'll do that.
I'm basing my plan mainly on this:
"11. If I lose my results letter, can I get a replacement?
No. Your results letter is an important document. Please keep it safe as we will not replace it if you lose or damage it. If you lose your results letter, please contact the Home Office."
It does say that you can't get a replacement. However, it also says to contact the Home Office. Why should you contact the Home Office if the only remedy is to simply retake the test and get a new pass certificate? The Home Office can't issue you a new letter, and they can't re-test you. It follows that you contact the Home Office because the testing center transmits your test information to them and then destroys it themselves, so only the Home Office has your test ID.
It also seems absurd (although absurdity in the immigration process is nothing new to any of us, so that may not mean anything) that the flimsy ink-jet paper they give you is your only evidence of the £50 test you just took. I showed them my BRP at the testing center and they linked the test to me with it. The test is also linked to my Home Office reference number, because that's what UKVI used to get me my test ID number. So they'll be able to see electronic records that I've passed the test when they're assessing my application, even if they don't have a piece of paper saying so.
And I'll definitely post again when the application is finally processed. I hope it goes well.