Free Traffic Legal Advice

National Highways are responsible for motorways and should have a record of temporary limits imposed

As your wife appears to be the only possible driver, and the police have refused to offer a course (to which there is no legal entitlement anyway)  the fixed penalty seems to be the best available outcome, she cannot get less than 3 points if she goes to court and the financial penalty will be greater. 3 points is unlikely to have much effect on her insurance premium, if any.

"24MPH is 10% + 2MPH of 20 speed limit."

That formula gives the speed at which the police will normally take action, not the speed you can do before they take action, if you have read otherwise elsewhere it is wrong, and a mistake often made. Just to make it clear, as soon as you exceed the limit you are committing the offence, but the police use their inherent discretion not to take action until a higher level.

The problem with ignoring the fixed penalty, so that you are charged, hopefuly under the wrong act and have to go to court is that it is a high risk strategy, if it fails you could end up seriously out of pocket financially and no one can tell you your chances of success, but the court legal advisor may well not be clued up on the obscurities of this legislation and will be unlikely to be sympathetic to a legalistic defence if you are charged under the wrong legislation and you will be arguing a legal point before him/her. If you are charged under the correct legislation you will have to argue the point made by the policeman, or plead guilty. While it is interesting for the posters here, I think the smart thing is to accept the fixed penalty.

I have had a clean licence my whole life and feel its unfair for me to get penalised for something like this?

The thing is, you are not being assessed for the standard of your driving over your whole life or even over this journey, but for a very brief period when this incident occurred. You know that you should not have overtaken on the inside, even if the other party's driving was erratic and provocative. By all means point this out as mitigation, but if the police persist, you would be wiser to accept an out of court offer than defend a charge in court, because if you do that and lose the consequences will be considerably more serious. In a situation where there is fault on both sides, it tends to be the party who first complains to the police who wins, unfortunately.

Is it just better if I get sjnp to just respond guilty, or do mitigation ?

it is all too easy to write what you think is mitigation but actually makes things worse, in your case saying that you forgot that there might be fireworks in early November, and you were so distracted by them that you forgot that you were in the course of slowing for a red light and went straight through it, is unlikely to impress the court. When you have no useful mitigation, do not try to put something forward, simply plead guilty.

Is THIS the place where the incident occurred? If so traffic in lane 1 needs to allow traffic from lane 2 to filter in, which may be difficult in heavy traffic. It is obviously difficult to really judge who was at fault here without a video, and bear in mind that the officer may have a video and be able to produce it in court. Realistically the court is likely to prefer the officer's version of events if you take this to court, and that is your only alternative to accepting the fixed penalty. If you take it to court and lose it will cost you much more than £100, there will be an income based fine and prosecution costs of over £600 and possibly more than 3 points. Although it will stick in your throat as you feel you were in the right, I would say that the sensible thing to do here is to accept the fixed penalty.
This has been voted the worst junction in Britain for good reason.