Only email if that's one of the options for appealing - the back of the notice should tell you what methods they accept.
If an occupant was a blue badge holder then make this clear and perhaps include a photo of the badge. Make clear that the driver entered their reg into the tablet and that their system appears to have failed.
You might be wise to appeal as the keeper, not revealing who was driving... You could then include something about their failure to comply with PoFA to hold the keeper liable.
I note from your correspondence that you are not seeking to hold me liable as the registered keeper, under Schedule 4 of the Protection of Freedoms Act 2012. There is no obligation for me to name the driver and I will not be doing so. No reasonable assumptions can be drawn as to the identity of the driver.
Feel free to show us a draft before submitting anything.