These "squeeze" type restrictions are, to use a Dickens expression, a "London particular" so can be traps to the non-london person. Camrose Avenue has been a regular on this forum for quite some years, and, sorry to have to say it, you will find it very difficult to overturn the PCN. Having said that, there is no harm in trying, so prepare a draf of your representation based on your narrative here, and post it for review. It would seem weather is the key issue, but your non-familiarity with the London traps is another, and could get them to cancel the PCN if you work the reps the right way. So best to be non-confrontational.
Of course, the real question is this, - if the width restriction is to stop heavy and wide vehicles from using the road, why do the councils issue PCNs to vehicles that are NOT heavy and wide ? Of course we all know the answer, it's their Magic Money Tree gifted by Parliament.