You might solve the incognito detection problem but you open up the opportunity for notification-fatigue if users end up getting lots of these permission requests.

Plus it wouldnt get around the root problem, which I assume to believe is

"you have reached your monthly limit of articles, please pay"

opens private tab to read article

"we see you are in a private window, please load our site in a normal window"

browser perfectly mimics regular mode but now you need to grant permission

"we see you havent granted permission, please allow access to read our article"