There is only one charge - it will be under section 87 (1) of Road Traffic Act 1988 - the learner driver was driving otherwise than in accordance with their licence. It sounds like the attached police statement refers to lack of L plates and lack of supervising driver. Either or both of these make the driver guilty of the offence. If the lack of supervising driver is disputed, the learner driver is still guilty of the offence. Dragging it out at court may only serve to increase costs to the defendant.
As you have already been advised, the reason for stopping the vehicle is irrelevant as the police are permitted to stop any vehicle under the Road Traffic Act. Your complaint regarding the subsequent search is also irrelevant to the charge.
Again as advised above, if your friend accepts driving without L plates displayed, he should plead guilty. I don't think it will make any difference to the fine that he in fact had a supervising driver. If the court thinks it will make a material difference to sentence, then the court can hold a Newton Hearing to hear evidence on this. I cannot imagine they will go to the time and effort and will probably just accept what your friend says.
Careful not to push too much for technicalities - as the supervising driver, technically you are guilty of causing and permitting the offence - also punishable with 3PP and a fine.
Having asked for free advice, perhaps you should accept what several contributors have now told you.