The V5 must be registered to the registered keepers name and address, otherwise you'd never get your tax reminder. Common sense surely dictates that much?
Also refusing to go into the BiB car is just going to cause you to instantly fail the attitude test and isn't worth the bother. Even if there is only one of them they can make up any old story that they want, why would it be any different if it was two? It's actually more awkward with two of you as one would be relying on the other one to back him up and not drop him in it
They should have no problem with you recording everything. If I were genuinely in the right then I'd ask them to write everything down on their notepad and we'd both sign it, if I didn't have a recording device on me. Again though, we're back to failing the attitude test in a big way.
£30 is peanuts, consider it a lesson learnt to always have your paperwork in order.