I've dug through every notification setting I can find in iOS, and unfortunately there's nothing related to only allowing one notification. In previous versions of iOS, there were actual options to limit the number of notifications you'd receive from an App, but it looks like those have been removed in iOS 9

I believe at this point the best option would be to turn Do Not Disturb mode on after cc posts in the thread, as you'll already be watching the thread F5ing, and really the only notification you'd be looking for is cc's post anyways. Unfortunately this won't really help if he opens the even immediately after posting (unless you skip using Tapatalk and instead use Safari after getting the notification).
It's definitely a janky solution, but it should work (unless Tapatalk ignores DND for some reason). Also, make sure that you go into Settings > Do Not Disturb and set the 'Silence' field to Always, otherwise it'll only mute the notifications when your phone is locked.

Android users, if you do have the Do Not Disturb mode (I know Android M does, not sure about earlier versions), make sure you select 'Total Silence'.
Other ideas (that aren't guaranteed to increase your chances at winning, but hopefully it helps

):
- Install another mobile browser just for Clackvent, and set the homepage to https://geekhack.org/index.php?action=post;topic=77786.0 That way when you open the browser, it's already on the reply page.
- Follow clickclack and ClackFactory accounts in Tapatalk. I'm not sure if it notifies you when they post, but it's worth a shot I guess, right?