As you correctly observe, they are making no attempt to hold you liable as the registered keeper, and as such, I'd recommend a simple appeal along the lines of the below:
Dear Sirs,
I have received your Parking Charge Notice (Ref: ________) for vehicle registration mark ____ ___, in which you allege that the driver is liable for a parking charge. 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 ("The Act"). You have chosen not to issue a Notice to Keeper in accordance with The Act, and it is now too late for you to do so.
There is no obligation for me to name the driver and I will not be doing so - please do not write to me asking me to name the driver. I am unable to help you further with this matter, and look forward to your confirmation that the charge has been cancelled. If you choose to decline this appeal, you must issue a POPLA code.
Yours,
You must appeal each charge separately, so 2 appeals. If they do not respond, chase them. I wouldn't raise any of the other issues you mentioned at this stage, as it potentially over-complicates a simple matter, and I don't think your signage point (about it not being clear it's a disabled bay) would be particularly convincing based on the photos you've showed us.