IMO, you are seeking to conflate two issues such that the one, having to pay a fee for renewing the permit and the lack of notice, in some way excuses parking without a permit and is a defence against the PCN.
IMO, they are separate issues.
The permit terms are clear: it's the holder's responsibility.
On which point, I refer to this comment in their letter:
In your correspondence, you have highlighted the following:
1. The expired permit was issued for my mother, a Blue Badge holder, whose ability to park near our residence has been seriously impacted.
So, who is the registered keeper of the vehicle in question? You say the NTO is addressed to you and your first post states 'my permit', so what has your mother got to do with this?
IMO, the registered keeper does not have the basis of a successful appeal.