IOS – Way to temporarily disable badge notifications while using iPhone

There might be another app that will work, eNotify. While I haven't tried it there is a lite version you can use to see if it works for you. This is along the think different solution. Send the email to a special email account and get annoying alerts.

eNotify provides direct local notification for incoming messages along with custom email alert sounds for different accounts, senders, subjects, and recipient addresses. Easily silence background email noise and stay aware of important message traffic! In addition, the message sender and subject can be shown via local notice, so you'll always know when your spouse wants you to drop by the store on the way home, without unlocking your phone!

(Since the accepted answer for this question is outdated and the actual answer is buried as a comment, I’m providing a better answer for all future people with this question, in hopes that it will become more visible.)

The default behavior for the Do Not Disturb feature is stopping notifications while the screen is locked, so that your iPad/iPhone won’t keep buzzing in your pocket/backpack/desk while you’re trying to concentrate on something else.

That’s good, but not what we want. We want to stop notifications while we’re using the device.

Well, at least since iOS 12 there’s an additional setting within Settings > Do Not Disturb that you can turn on, here:

All you need to do is change the “Silence” option to “Always” in this screen. It will keep any and all notifications away from your screen as you write, read, present content to others, play a game, etc.