As Chris has said, if the Magistrates believe your guilty plea is “equivocal” (that is, not a clear acceptance of your guilt but one made, perhaps, for the sake of convenience) they are very likely to refuse to accept it and instead enter a not guilty plea and order the matter to go to trial. In writing to the Bench as you suggest they will certainly gain this impression.
As far as the trial goes, it is not for the defendant to prove anything. The prosecution (in this case the DVLA) has to prove the issue beyond reasonable doubt. In cases such as this they have to prove a negative – that is, that a licence was not held, or a test was not passed. I’m sure they’re used to it otherwise anybody who had not passed a test could insist that at some time they had held a full licence. However, it will be for the Magistrates to decide based on the evidence presented on the day. The DVLA will simply insist their records are infallible. In his defence your friend should be prepared to produce some evidence – even if it is only verbal – that he had passed a test.
An Absolute Discharge is unlikely to be awarded in the case of a conviction. If the Magistrates believe your friend's account in preference to that of the DVLA they must acquit him. In any case an Absolute Discharge is simply a sentencing option and a conviction will still be recorded. Also any other appropriate orders such as penalty points or a ban will still be imposed.