Clearly you parked the previous day in the evening, because the PCN was served on or after 14.52, on the 25th November. Looking at the photos, it's difficult to see what you could put into representations. I would suggest you major on what you have told us, namely dark, raining, baby to deal with, (take out pram/buggy, take baby out of car, and put in buggy, close and lock car) all the while the rain falling relentlessly. Post up their inevitable rejection when you get it as they may shoot themselves in the foot in it, (although this is not common). They'll reoffer the discount so don't worry on that score.
Are you the owner of the car as per the V5C registration certificate, and is it up-to-date ? If the matter goes further they will send out a Notice to Owner by post to the owner registered on the V5C
