I'm only really familiar with android and I know that Google cache the purchases on the user's device so it should work while offline.
That's correct what you wrote in the last post if I'm understanding it correctly, you can use a variable to toggle something when you receive info about the purchases, not to track whether they have purchased something on a long term basis.
I know it's not good to use local storage to track purchases as people can swap devices and then the purchases are missing. For the banner I guess design the events in a way that you don't start showing banners until you've checked for the purchase.