Hi there Snowdon,
I told you a lie earlier. While uploading documents this weekend I noticed that the bill I used for the Life in the UK test actually does have my full name, spelled out just like it is on my BRP. So that's probably why they barely glanced at it, and if you want it to be a non-issue, probably a really good idea to take the time to get the bill name adjusted!
I'd also say - and your mileage may vary - but because I picked a half-term week to take the test, the time slots filled up really quickly. Obvs that may have just been the testing centre I was planning to use, but, just thought I'd suggest that you shouldn't leave it until the last minute to book the test.
And, again, wishing you all the best of luck when you take the test!