I agree that producing your documents at a police station and getting some kind of proof is an excellent way to nullify any attempt to prosecute you for this offence.
I'm afraid I don't get the rest of it about asking them to prove you contracted with them (I know I'm a bit thick
) your evidence should eliminate any court case to do with producing documents. As to if you will be reported for not having L plates, my guess is if he was a normal officer then no you won't as he will almost certainly have more pressing things to do (like citing up old ladies! I'M JOKING
) If he was a traffic cop then I would expect him to report you as this is his bread and butter (to use their phrase.) It is one of those things that is hard to predict.
If I was you though I would put L plates on ASAP, as I have known police officers to pull the person again and summon them for the offence. Almost certainly the CPS will put the two summons together in the one court case, which means two sets of officers who have (allegedly) seen you commit the offence and if you are found guilty that (potentially) is four counts which is 4 x £60 fine and 4 x 3 points.
That is just what I would do