It looks like the solution to this problem may be to add the application to the home screen. It seems that the author of the blog post that you referenced added clarification on this, after it was originally published. If you scroll down to the section 'A Note On Web Applications Added to the Home Screen', it reads:
As mentioned, the seven-day cap on script-writable storage is gated on
“after seven days of Safari use without user interaction on the site.”
That is the case in Safari. Web applications added to the home screen
are not part of Safari and thus have their own counter of days of use.
Their days of use will match actual use of the web application which
resets the timer. We do not expect the first-party in such a web
application to have its website data deleted.
If your web application does experience website data deletion, please
let us know since we would consider it a serious bug. It is not the
intention of Intelligent Tracking Prevention to delete website data
for first parties in web applications.
Apple: Relax, we're not totally screwing web apps. But yes, third-party cookies are toast
Sure, we'll delete local data after seven days but there's a way to avoid that
Last but not least, I hope you are using the Web Crypto API to store your private keys securely. See https://crypto.stackexchange.com/questions/35530/where-and-how-to-store-private-keys-in-web-applications-for-private-messaging-wi/52488#52488 for more info.