Hello Sterling, thanks for the thorough reply. I read through the article you cited and double checked my settings against it, everything seems to be in order there. I haven't yet found a solution to my problem though, unfortunately.
You're correct about my setup. I'm the Family Organizer. I've got my under-13 son on the account, Ask-To-Buy enabled by default, and I double checked under Settings > iCloud > Family > Member that the Ask-To-Buy switch is indeed enabled.
So, order of events: Let's say he puts in a buy request for Random App. Within a few seconds I'll receive the "XYZ Child wants to buy Random App, Approve/Deny" banner on my phone/pad, which will generally be locked and holstered at the time. If I happen to hear the alert, and manage to get my phone into my hand while the alert is still visible on the lock screen, I can then swipe right on the banner, approve or deny, everything's groovy. However, if I see the alert but don't manage to execute the swipe before the 30 seconds, or whatever the interval is, has past then banner auto-hides itself and seems to disappear.
When I swipe down to expose the Today/Notifications screen, the request isn't there. Even if I check it immediately after failing to catch the banner in time. At most I've missed it by minutes, never 24 hours or more.
I checked Settings > Notifications, there doesn't seem to be an item for 'Family' or 'iCloud' but notifications are enabled for 'App Store' and 'iTunes Store', if that's relevant.
Also, I should have mentioned initially, I'm used an iPhone 6 and iPad Air 2, both running iOS 8.3. He's using an iPad Mini 3, iOS 8.3.
Thanks again for your help!
-Jason