I'm sorry for your situation as I know that they can be very officious.
This is almost certainly a vehicle defect allegation.
What you have received is a conditional fixed penalty offer. You do not have to accept that. It is just an offer affectively to divert it away from court.
The risk of refusing is that they may summons you. I am not sure whether you dispute that the tyre was deflated.
I do realise that they were looking for an excuse to find something but the problem is that they have. It isn't a defence to a charge to say that the police were unreasonable even though I know that they can be.
If you tell me whether or not you accept the tyre was flat or deflated then it will give us some pointers over whether you have a defence or not
you do also have the option to write to them asking them to consider the merits of this prosecution. He should probably do that. They can always review this for public interest.
However, obviously you cannot rely upon the outcome being in your favour.
Can I clarify anything for you?